On the p-adic Beilinson conjecture for number fields

ON THE p-ADIC BEILINSON CONJECTURE F OR NUMBER FIELDS. AMNON BESSER, P A UL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT De dic ated to Je an-Pierr e Serr e o n the o c c asion of his eightieth birthday. Abstract. W e formulate a conjectural p -adic analogue of Borel’s theorem re- lating regulators for higher K -groups of num b er ﬁelds to sp ecial v alues of the corresponding ζ -funct ions, using syntomic regulat ors and p -adic L -functions. W e also formulat e a corresponding conjecture for Artin motiv es, and state a conjecture ab out the precise relation betw een the p -adic and classical situa- tions. Pa rts of the conjectures are prov ed when the n umber ﬁeld (or Artin motiv e) is Abeli an ov er the r ationals, and al l conjectures are v eriﬁed nu meri- cally in some other cases. 1. Introduction The B e ilinson conjectur e s ab out sp ecial v alues of L -functions [2] are a far rea ch- ing gener alization o f the c la ss num b er formula fo r the Dedekind zeta function. F or every algebraic v a riety X over the rationals it predicts the lea ding term of the T a ylor expansions of L ( H i ( X ) , s ) at certain points, up to a ra tional m ultiple, in terms of a rithmetic infor mation asso ciated with X , na mely , its alg ebraic K -groups K j ( X ) [43]. More g enerally , these c o njectures can also b e formulated for motives. There have been several important steps taken to wards v eriﬁcatio n o f these co n- jectures in v ar ious ca ses, although, strictly sp ea king they hav e only been v eriﬁed completely in the case where X is the sp ectrum of a num be r ﬁeld, wher e they follow from famous theorems of Borel [10, 11]. T o motiv ate what follo ws, let us brieﬂy recall the conjecture that in terests us the mos t (for a n intro duction see [23, 4 6]). O ne asso ciates with X tw o cohomolog y groups. The ﬁrst one is the Delig ne cohomolog y H i D ( X / R , R ( n )), which is an R - vector space . The second is “integral” motivic c o homology H i M ( X / Z , Q ( n )), which may b e deﬁned as a cer tain subspace of K 2 n − i ( X ) ⊗ Z Q . There is a reg ulator map deﬁned by Beilinson, (1.1) H i M ( X / Z , Q ( n )) → H i D ( X / R , R ( n )) . If 2 n > i + 1 then de t H i D ( X / R , R ( n )) has a rational structur e coming from the relations betw een H i D ( X R , R ( n )) and the de Rham and singular cohomolo gy gr oups of X [4 6, p.3 0]. The ﬁrst pa rt of the conjecture is that the map in (1.1) induces an isomo rphism betw een the lefthand side tenso red with R and the r ighthand side, and conseque ntly 2000 Mathematics Subje ct Classiﬁc ation. Primary: 19F27; secondary: 11G55, 11R42, 11R70. Key wor ds and phr ases. Beilinson conjecture, Borel’s theorem, num ber ﬁeld, Artin m otiv e, syn tomic regulator, p -adic L -function. 1 2 AMNON BES SER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS R OBLOT provides a se c ond ratio nal structur e o n det H i D ( X / R , R ( n )). The second part of the conjecture states that, assuming a suitable functional equation for L ( H i − 1 ( X ) , s ), these tw o ra tional structur es diﬀer from each other b y the leading term in the T a ylor expansion of L ( H i − 1 ( X ) , s ) at s = i − n . Because of the exp ected functional equation one can reformulate the conjecture in terms of L ( H i − 1 ( X ) , n ) (see [2, Corollar y 3.6.2] or [31, 4.12 ]). As mentioned b efore, this conjecture has only b een v eriﬁed in the case of num b e r ﬁelds, due to diﬃculties in the computation o f motivic co homology . What has b een veriﬁed in several other cases is a form of the conjecture in which one assumes the ﬁrs t par t. F or this one ﬁnds dim R H i D ( X / R , R ( n )) ele ments o f H i M ( X / Z , Q ( n )), chec ks tha t their images under (1.1) a re indep endent , hence should form a basis o f H i M ( X / Z , Q ( n )) according to the ﬁrst par t of the conjecture, and veriﬁes the second part using these elements. The idea that ther e should be a p -adic a nalogue of Beilinson’s conjectures has bee n around since the la te 80’s. Such a conjecture was formulated a nd proved by Gros in [29, 30] in the case o f Artin mo tives asso ciated with Dirichlet characters. In the weak sense men tioned befor e, it was pr ov ed for certain CM elliptic curves in [1 7] (where the relation with the syntomic regulato r is proved in [4] and fur- ther elucidated in [7]), and for e lliptic modular for ms it follows from K ato’s work (see [4 7]). The b o ok [42] contains a very gener al conjecture ab out the existence and pr op er- ties of p - a dic L -functions, fro m which one can derive a p -a dic B e ilinson conjecture. Rather than explain this in full detail we sha ll give a sketch o f this conjecture similar to the sketc h ab ove of the Beilinson conjecture. F or the p -adic Beilinso n conjectures o ne has to repla ce Deligne cohomolog y w ith syntomic cohomolo gy [30, 41, 3], the Beilinson regulator with the sy n tomic r egula- tor, and L -functions with p -adic L - functions . Synt omic coho mo logy H i syn ( Y , n ) is deﬁned for a smo o th scheme Y of ﬁnite type ov er a complete discrete v aluation ring of mixed characteristic (0 , p ) with p erfect residue ﬁeld. F or the Q -v a riety X w e obtain, for all but ﬁnitely many primes, a map (1.2) H i M ( X / Z , Q ( n )) → H i syn ( Y , n ) where Y is a smo o th mo del for X over Z p . This cohomolo gy gr oup is a Q p -vector space. The theory of p -adic L -functions star ts with Kubo ta and Leopo ldt’s p -adic ζ - function [35], ζ p ( s ), which is deﬁned b y interpolating s pe c ial v alues of complex v alued Diric hlet L -functions. This principle has b een extended to ζ - a nd L - functions in v a rious situations, resulting in corresp onding p -adic functions for tota lly real nu mber ﬁelds [1, 15, 22], CM ﬁelds [3 2, 33] and mo dular forms [37]. (Given the o ccasion, let us note that the appro ach of Deligne and Rib et using mo dular forms was initiated b y Serr e [48].) The p -adic Beilins on conjecture therefore ha s many similarities with its co mplex counterpart. How ever, there is a very impo rtant diﬀerence. In g e neral, when 2 n > i + 1 , there is no hop e that (1.2) induces a n isomor phism after tenso r ing the lefthand side with Q p . T o see this, cons ider a num ber ﬁeld k with ring of alge braic integers O k . By Bo rel’s theo rem (see Theorem 2 .2) we hav e, in acc ordance with Be ilins on’s ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 3 conjectures, dim Q H 1 M (Spec ( k ) / Z , Q ( n )) =  r 2 when n ≥ 2 is even r 1 + r 2 when n ≥ 2 is o dd with r 1 (resp. 2 r 2 ) the n umber of real (re s p. complex) em b eddings of k . Thu s, motivic cohomology “k nows” ab out the num b er of real and complex em b eddings of k , as does, by its deﬁnition, Deligne cohomolo gy , which in this case b ecomes H 1 D (Spec ( k ) / R , R ( n )) ∼ = { ( x σ ) σ : k → C in R ( n − 1) r 1 +2 r 2 such that x σ = x σ } . But syntomic coho mology , which dep e nds only on the completion of k at p , do es not. In this c a se, we obtain H 1 syn (Spec( O k ⊗ Z Z p ) , n ) ∼ = Q r 1 +2 r 2 p . The so lution to this pr oblem s uggested in [42] is to make the p -adic L -function depe nd on a subspace of syn tomic cohomolog y whic h is complementary to the image o f the regula tor. While this might seem artiﬁcial, there are other reas ons for choosing this so lution. In mos t ca ses one choos es a particular subs pa ce and obtains a s pe c ial c a se of the conjecture. In the imp ortant sp ecial c ase o f a totally r eal num b er ﬁeld, or more genera lly an Artin motive over Q , asso ciated with a Galois representation whose k ernel is totally real (let us ca ll these totally re a l Artin motives), no such subspa ce is requir ed when n ≥ 2 is odd (see P rop osition 3.12). The sa me holds for Artin motives where the conjugacy cla ss of complex conjugation acts a s multiplication b y − 1 and n ≥ 2 is even (w e may think of thos e a s the “ negative part” of CM Artin motives). If χ denotes the character asso ciated with either representation, then the Beilin- son conjecture re la tes the regula to r of K 2 n − 1 with the Artin L -function of χ at n . F o r the p -adic L -functions one has to co nsider χ ⊗ ω 1 − n p with ω p the T eichm¨ uller character for the prime n um b er p . Then the ﬁxed ﬁeld of the kernel of the rep- resentation underlying χ ⊗ ω 1 − n p is totally real, and it is perhaps no co incidence that in precis ely this case the existence of a p - adic L - function that is not ide nticaly zero has b een established, by [1, 15, 2 2] for the ca se of ﬁelds and by [2 8] for Artin motives. The g oal of the pres ent work is to descr ib e in detail the co njectures for the cases of totally rea l ﬁelds, totally real Artin motives, as w ell as the “negative part” of CM Artin motiv es, and de s crib e the (conjectural) rela tion b etw een the cla ssical a nd p -adic conjectures. W e test everything numerically in s e veral ca ses, and als o deduce most o f the conjectures for Ab elian Artin mo tives fro m w ork by Coleman [16]. There hav e b een several developmen ts that allow us to ca rry o ut this numerical veriﬁcation. In [19] de Jeu prov ed part o f Za gier’s c onjecture concerning a more explicit description of the K -theory (tensor ed with Q ) of n umber ﬁelds (see Sec- tion 4). While this conjecture is not k nown to give the K-theor y o f suc h ﬁelds in all ca ses, it does in practice. Th us it provides a way of computing them, and Paul Buckingham wr ote a c o mputer implemen tation for this. In [5] Besser and de Jeu computed the synt omic r egulator for (essentially) the part of the K -theory of a num b er ﬁeld describ ed by Zagier’s conjecture and show ed that it is g iven by applying the p -adic poly lo garithm. Those p -adic po ly logarithms w ere inv en ted by Coleman [1 6] using his theory of p -adic integration but are not s o eas y to compute. In [6] Bes ser and de Jeu devised an algo r ithm for this computation. T aken tog ether, these developments allow us to compute (1.1) and (1.2) for num ber ﬁelds. Fina lly , building on e a rlier work in [45], Roblot has dealt with the computational a sp ects 4 AMNON BES SER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIE R-FRANC ¸ OIS ROB LOT of co mputing p -a dic L -functions for Ab elian characters ov er Q or a real quadra tic ﬁeld [44 ]. This pa pe r is organized a s follows. In Section 2 we recall Bor el’s theorem a s well as v ar ious facts ab out L -functions and p -adic L -functions, and fo r mulate a conjectural p -adic a nalogue of Bor el’s theorem. In Section 3 w e introduce Artin motives with co eﬃcients in a num ber ﬁeld E in ter ms of r epresentations of the Ga- lois gr oup, deter mine when the Q -dimension of the left-ha nd side of (1.2) equals the Q p -dimension of the rig ht -hand side (corresp o nding to equality in Prop osition 3 .12), deﬁne both classical a nd p -adic L -functions with co eﬃcients in E , and formulate the motivic Beilinson conjecture with co eﬃcients in E , Conjecture 3 .18, a small part o f which is the same as a conjecture by Gro ss. In Sec tion 4 we describ e the set- up for ﬁnding elemen ts in the K -g roups o f n umber ﬁelds using Zagier’s conjecture, and the cla ssical a nd p -adic regulato rs on them. W e a lso prove most o f Co njectur e 3 .18 for Ab elian Artin motives ov er Q (Prop osition 4.17 and Remark 4.18). In Sectio n 5 we dis c us s a few computational asp ects o f implementing Zagier ’s conjecture and describ e the Artin motives that we consider la ter for the numerical examples, and in the pro ces s prov e Gros s’s conjecture for Artin motives o btained fr o m S 3 and D 8 -extensions of Q . Then in Section 6 w e sketch how the p -adic L -functions can be computed in certain case s, a nd make the requir ed Br auer induction explicit for the Artin motives we wan t to consider. Fina lly , the last section is devoted to the results of the numerical calculatio ns for examples. Amnon Bes ser, Paul Buckingham a nd Rob de Jeu would lik e to thank the EC net work Arithmetic Algebraic Geo metr y for trav el supp ort. Rob de Jeu would like to thank the T ata Institute o f F unda ment al Res e arch for a pro ductive stay dur ing which this pap er was w orked on, and the Nuﬃeld F oundation for a g r ant under the Underg r aduate Resea rch Bursary programme (NUF-URB03) that enabled Paul Buckingham to develop a computer progra m for ﬁnding the necessary elements in K - theory . Amnon Bes ser would like to tha nk the Institute for Adv a nc e d Study in P rinceton for a s tay during which part of the pa p er w as work ed on, and the Bell companie s F ellowship and the Ja mes D. W olfensohn fund for ﬁnancial s upp or t while at the institute. Finally , the authors w ould like to thank Alfred W eiss for very use ful expla na tions a b out p -adic L -functions, and, in particular, for bringing to their attent ion the pap er [28]. Notation Throughout the pap er , for an Ab elian group A we let A Q denote A ⊗ Z Q . If B is a subgroup of C a nd n an integer then we let B ( n ) = (2 π i ) n B ⊆ C W e normalize the absolute v alue | · | p on the ﬁeld of p -adic n umbers Q p in such a wa y that | p | p = p − 1 , and use the same notation for its extensions to an algebr aic closure Q p and C p = ˆ Q p . 2. The p -adic Beil ins o n conjecture f o r tot all y real fields Let k b e a n um b er ﬁeld with r 1 real em beddings , 2 r 2 complex em b eddings, ring of algebraic integers O k , and dis criminant D k . As is well-kno wn, O ∗ k is a ﬁnitely generated Abelian group of ra nk r = r 1 + r 2 − 1 , a nd its regulator R satisﬁes w p | D k | Res s =1 ζ k ( s ) = 2 r 1 (2 π ) r 2 | Cl( O k ) | R , with Cl( O k ) the c la ss gr oup of O k , and w the n um b er of r o ots of unity in k . Becaus e K 0 ( O k ) ∼ = Cl( O k ) ⊕ Z and K 1 ( O k ) ∼ = O ∗ k , so | Cl( O k ) | = | K 0 ( O k ) tor | and w = | K 1 ( O k ) tor | , this can be int erpre ted as a statement ab out the K -theory of O k , and it is from this p oint ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 5 of v ie w that it can be genera lized to ζ k ( n ) for n ≥ 2. Namely , in [43], Quillen prov ed that K m ( O k ) is a ﬁnitely generated Ab elian g r oup fo r a ll m . B orel in [10] computed its ra nk when m ≥ 2. F or m even this ra nk is zero, but for o dd m it is r 1 + r 2 or r 2 , and in [1 1] he show ed that a suitably deﬁned reg ulator o f K 2 n − 1 ( O k ) is related to ζ k ( n ). Since K 2 n − 1 ( O k ) ∼ → K 2 n − 1 ( k ) when n ≥ 2 we can rephrase his results for K 2 n − 1 ( O k ) in terms o f K 2 n − 1 ( k ). Also, we replace Borel’s reg ulator map reg B : K 2 n − 1 ( C ) → R ( n − 1) ( n ≥ 2 ) with Beilinson’s regulato r ma p reg ∞ (see [46, § 4]), whic h is half the Borel ma p r egulator b y [12, Theorem 10 .9 ]. Beca us e k ⊗ Q C ∼ = ⊕ σ : k → C C , and n > dim Spec( k ⊗ C ), we obtain by [46, p.9] (2.1) H 1 D (Spec ( k ⊗ C ) / R , R ( n )) ∼ = H 0 (Spec ( k ⊗ C ) / R , R ( n − 1)) ∼ =  M σ : k → C R ( n − 1)  + , which co nsists of those ( x σ ) σ with x σ = x σ . Finally , for any embedding σ : k → C we le t σ ∗ : K 2 n − 1 ( k ) → K 2 n − 1 ( C ) be the induced map. Theorem 2.2. (Borel ) L et k b e a numb er ﬁeld of de gr e e d , with r 1 r e al emb e ddings and 2 r 2 c omplex emb e ddings, and let n ≥ 2 . Then the r ank m n of K 2 n − 1 ( k ) e quals r 2 if n is even and r 1 + r 2 if n is o dd. Mor e over, t he map (2.3) K 2 n − 1 ( k ) → M σ : k → C R ( n − 1) α 7→ (reg ∞ ◦ σ ∗ ( α )) σ emb e ds K 2 n − 1 ( k ) / torsio n as a lattic e in ( ⊕ σ R ( n − 1)) + ∼ = R m n , and the volume V n ( k ) of a fun damental domain of this lattic e satisﬁes (2.4) ζ k ( n ) p | D k | = q π n ( d − m n ) V n ( k ) for some q in Q ∗ . Remark 2.5. In Theor em 2 .2 m n = 0 precis ely when k is totally real and n ≥ 2 is o dd. In this case the given rela tion holds (with V n ( k ) = 1) by the Siege l-Klingen theorem [40, Chapter VI I, Coro llary 9.9]. This theorem is equiv alent with B e ilinson’s conjecture for k . In order to deal with this in detail (see Remarks 2.20 and 3.24) and in order to introduce p -adic L -functions we rec a ll some facts abo ut Artin L -functions [40, Chapter VII, § 10-12 ]. Let k be a num b er ﬁeld, d = [ k : Q ], and let χ be a C -v alued Artin character o f Gal( k /k ). F or a prime num ber ℓ we deﬁne Eul ℓ ( s, χ, k ) = Y l | ℓ Eul l ( s, χ, k ) where E ul l ( s, χ, k ) is the recipr o cal of the Euler factor for l a nd the pro duct is ov er primes l of k lying a bove ℓ . Then for s ∈ C with Re ( s ) > 1 we ca n write the Artin L -function of χ as L ( s, χ, k ) = Y ℓ Eul ℓ ( s, χ, k ) − 1 . F o r an inﬁnite place v of k we let L v ( s, χ, k ) = ( L C ( s ) χ (1) when v is complex, L R ( s ) n + L R ( s + 1) n − when v is real, 6 AMNON BES SER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIE R-FRANC ¸ OIS ROB LOT where L R ( s ) = π − s/ 2 Γ( s/ 2), L C ( s ) = 2(2 π ) − s Γ( s ) and n ± = d 2 χ (1) ± P v 1 2 χ (F r w ), with the sum taken ov er the re a l places v of k and F r w the gener ator of the ima ge of Ga l( C / R ) in Ga l( k /k ) co rresp onding to any e x tension w : k → C o f v . Then Λ( s, χ, k ) = L ( s, χ, k ) Y v | ∞ L v ( s, χ, k ) extends to a mero morphic function, satisfying the functional equation (cf. [40, p. 541]) Λ( s, χ, k ) = W ( χ ) C ( χ, k ) 1 / 2 C ( χ, k ) − s Λ(1 − s, χ, k ) with W ( χ ) a co nstant of a bsolute v a lue 1 and C ( χ, k ) = | D k | χ (1) Nm k/ Q ( f ( χ, k )) for f ( χ, k ) the Artin conductor of χ . Therefor e (2.6) L (1 − s, χ, k ) = W ( χ ) C ( χ, k ) s − 1 2 (2(2 π ) − s Γ( s )) dχ (1) × (cos( π s/ 2)) n + (sin( π s/ 2)) n − L ( s, χ, k ) . F o llowing [28, pp. 8 0–81 ] w e shall no w describ e a p -adic L -function L p ( s, χ, k ) when k is a totally real num ber ﬁeld, p a prime and χ : Gal( k /k ) → Q p a s uitable Artin character. W e b eg in with the case of 1-dimensio nal Ar tin characters. If σ : Q p → C is an y isomor phism then σ ◦ χ is a complex Artin character, so we hav e the Artin L -function L ( s, σ ◦ χ, k ). W e ma y a ls o view χ as a character o f a s uitable r ay cla ss group, s o by [4 0, Corollar y 9.9 and page 509 ] all L ( m, σ ◦ χ, k ) for m ∈ Z ≤ 0 are in Q ( σ ◦ χ ) a nd the v alues (2.7) L ∗ ( m, χ, k ) = σ − 1 ( L ( m, σ ◦ χ, k )) are indep endent of the choice of σ . In the same wa y , we deﬁne (2.8) Eul ∗ ℓ ( m, χ, k ) = σ − 1 (Eul ℓ ( m, σ ◦ χ, k )) , which is clea rly indep endent of the choice of σ . T o construct the p - adic L -function one ﬁnds a p -adic ana lytic or meromor phic function o n an o pen ball around 0 that int erp ola tes the v alues L ∗ ( m, χ, k ). F or 1-dimensional χ with the ﬁxed ﬁeld k χ of its kernel totally real this w as achieved indep endently by Deligne and Ribet [22], Barsky [1], a nd Cassou-Nogu` es [14, 15]. W e shall sketc h a pro o f of the follo wing theorem a nd Remark 2.13 below in Sec tio n 6. (When k χ is no t totally real the p -adic L -function is identically zero s ince the v alues interp o lated are all zero by the functional eq uation of the L -function.) Theorem 2.9 . F or p prime, let B in C p b e the op en b al l with c entr e 0 and r adius q p − 1 / ( p − 1) wher e q = p if p > 2 and q = 4 if p = 2 . If k is a total ly r e al numb er ﬁeld a nd χ : Gal( k /k ) → Q p a 1 -dimensional A rtin char acter, then ther e exists a unique C p -value d function L p ( s, χ, k ) on B satisfying the fol lowing pr op ert ies: (1) L p ( s, χ, k ) is analytic if χ is n on-trivial and mer omorphic with at most a simple p ole at s = 1 if χ is trival; (2) if m is a ne gative inte ger such that m ≡ 1 mo dulo ϕ ( q ) then L p ( m, χ, k ) = Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ) . ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 7 If χ : Gal( k /k ) → Q p is a ny Artin c haracter then by Brauer’s induction theo - rem [40, (10 .3)] there exist 1-dimensional Artin character s χ 1 , . . . , χ t on subgr o ups G 1 , . . . , G t of Ga l( k /k ) o f ﬁnite index, and integers a 1 , . . . , a t , such that (2.10) χ = t X i =1 a i Ind Gal( k/k ) G i ( χ i ) . If k χ is tota lly r eal then we can ass ume that the sa me holds for the ﬁxed ﬁe lds k i of the G i , and we deﬁne the p -adic L - function of χ b y (2.11) L p ( s, χ, k ) = t Y i =1 L p ( s, χ i , k i ) a i , which is a meromorphic function on B (see Section 6). F or any negative integer m ≡ 1 mo dulo ϕ ( q ) the v alue L p ( m, χ, k ) is deﬁned a nd equals Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ) by well-known prop erties of Artin L -functions (see [40, Prop. 10.4(iv)]), showing the function is independent of how we expres s χ as a sum of induced 1 -dimensional characters. Remark 2.12. In [28], Greenberg prov es that the Main Co njecture of Iw asawa theory implies the p -a dic Artin conjectur e , that is, that the p -a dic L -function of an Artin character χ is analytic on the open ball B if it do es not contain the trivia l character, and ha s at most a simple p o le at s = 1 otherwise. It therefore follows from the pro of of the Main Conjecture by Mazur and Wiles [38] for p o dd that L p ( n, χ, k ) for p 6 = 2 is deﬁned for all in tegers n 6 = 1. In particular, the v alues of the p - adic L -functions in Co njecture 2 .17 b elow exist by The o rem 2.9, and those in Conjecture 3.18 exist when p 6 = 2 but hav e to be assumed to exist when p = 2. Remark 2 . 13. Let W p ⊂ Z ∗ p be the subgr oup of ( p − 1)-th r o ots o f unit y if p is o dd and let W 2 = {± 1 } . The T eichm¨ uller character on Gal( k /k ) is deﬁned as the comp osition (2.14) ω p : Gal( k /k ) → Gal( Q / Q ) → Ga l( Q ( µ q ) / Q ) ∼ → ( Z /q Z ) ∗ ∼ → W p , where th e last map sends a n elemen t o f ( Z /q Z ) ∗ to the unique elemen t of W p to which it is c o ngruent modulo q . F or an Artin character χ : Gal( k /k ) → Q p and a n in teger l , χω l p is a lso an Artin character. If m ≤ 0 satisﬁes l + m ≡ 1 mo dulo ϕ ( q ) and either χ is 1- dimensional or k χ is totally real then L p ( m, χω l p , k ) = Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ). The p -adic Beilinson conjecture is going to predict the spe cial v alues of p -adic L -functions in terms of a p -adic r egulator. The r equired reg ulator is the syntomic regulator [3, Theorem 7.5]. Let F be a complete discretely v a lued ﬁeld of charac- teristic 0 with p erfect residue ﬁeld o f characteristic p and let X be a scheme that is smo oth a nd of ﬁnite type over the v alua tion r ing O F . Then the a b ove mentioned pap er asso cia tes to X its rigid s y nt omic coho mo logies H i syn ( X, n ), as well as syn- tomic regulator s (i.e., Cher n c hara cters) reg p : K 2 n − i ( X ) → H i syn ( X, n ). In this work, unlike in [3 ], w e shall need to c hange the base ﬁeld F . W e therefore prefer to denote the syntomic cohomolo g y by H i syn ( X/ O F , n ). F or the formulation of the conjecture the following basic fact is r equired. Lemma 2.15. We hav e H 1 syn (Spec ( O F ) / O F , n ) ∼ = F , for al l n > 0 and c onse- quently we have a syntomic r e gulator r eg p : K 2 n − 1 ( O F ) → F . F urthermor e, the 8 AMNON BES SER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIE R-FRANC ¸ OIS ROB LOT map reg p c ommu tes in the obvious way with ﬁnite ex tensions of ﬁelds and wi th automorphisms of such ﬁelds, pr ovide d t hat their r esidue ﬁelds ar e algebr aic over the prime ﬁeld. Pr o of. The ﬁrst claim follo ws fr om pa rt 3 of [3, Prop o s ition 8.6]. F o r the second claim we note that b y part 4 of the same prop osition w e have in th is case an isomorphism b etw een syntomic and mo diﬁed syntomic cohomolog y (the latter only exists under the additional assumption on the residue ﬁeld). The c o mpatibility with ﬁnite base changes no w follows fro m the same result for mo diﬁed syn tomic cohomolog y in [3, Propo sition 8.8]. The s ame holds for a utomorphisms, althoug h not explicitly sta ted in the ab ov e refer ence, since the r elev a nt base change re sults, e.g., Prop ositio n 8.6.4, hold for this type of ba se change a s well.  As a consequence of the lemma we can also , by a buse o f no ta tion, deﬁne for any complete discretely v alued subﬁeld F ⊂ Q p the regulator reg p : K 2 n − 1 ( O F ) → F . In [5], tw o o f the authors of the present work show ed ho w one can sometimes compute the map reg p by us ing p -adic p o lylogar ithms. W e now r e s trict o ur attention to a totally r eal num be r ﬁeld k with [ k : Q ] = d . Our goal will b e to for mulate a co njecture that is the p -adic analogue of Theor em 2.2 for K 2 n − 1 ( k ) with n ≥ 2. Since this K -group is torsion when n is even but has rank d w hen n is odd, w e only consider o dd n ≥ 2. In prepara tion for the more g e ne r al co nstruction that will follow in Section 3 let us ﬁr st r e formulate Theo rem 2.2 in this sp ecial cas e. Le t a 1 , . . . , a d form a Z - ba sis of O k and let σ ∞ 1 , . . . , σ ∞ d be the embeddings of k int o C . Then we deﬁne D 1 / 2 , ∞ k = det( σ ∞ i ( a j )), a real ro ot of the discriminan t of k . Similarly , if α 1 , . . . , α d form a Z -basis of K 2 n − 1 ( k ) / torsio n then we let R n, ∞ ( k ) = det(reg ∞ ◦ σ ∞ i ∗ ( α j )). Then D 1 / 2 , ∞ k and R n, ∞ ( k ) are well-deﬁned up to sign, and the relation in Theor em 2.2 in this ca se is equiv alent with (2.16) ζ k ( n ) D 1 / 2 , ∞ k = q ( n, k ) R n, ∞ ( k ) with q ( n, k ) in Q ∗ . Now let F ⊂ Q p be the to po logical closur e o f the Galois clos ure of k em bedded in Q p in an y w ay . If σ p 1 , . . . , σ p d are the em bedding s of k into F then D 1 / 2 ,p k = det( σ p i ( a j )) is a ro ot in F of the dis c r iminant o f k . F or σ : k → F an embedding we deno te the induced map K 2 n − 1 ( k ) → K 2 n − 1 ( F ) b y σ ∗ . Then w e deﬁne a p - adic regulator in F b y R n,p ( k ) = det(reg p ◦ σ p i ∗ ( α j )). Both D 1 / 2 ,p k and R n,p ( k ) are well-deﬁned up to sign. Remark 2.13 sugge s ts that the r ole o f ζ k ( n ) in a p -adic analog ue of Theo rem 2.2 should be pla yed by L p ( n, ω 1 − n p , k ) / Eul p ( n, k ) wher e ζ k ( s ) = Q l Eul l ( s, k ) − 1 for Re( s ) > 1, s o we ca n hop e that L p ( n, ω 1 − n p , k ) D 1 / 2 ,p k = q p ( n, k ) E ul p ( n, k ) R n,p ( k ) for some q p ( n, k ) in Q ∗ . More precisely , be cause R n, ∞ ( k ) /D 1 / 2 , ∞ k and R n,p ( k ) /D 1 / 2 ,p k are inv ariant under reorder ing the σ ∞ i or σ p i , and transform in the sa me way if we change the bases of O k and K 2 n − 1 ( k ) / torsio n, we can make the following conjecture. Conjecture 2.17. F or k a total ly r e al numb er ﬁeld, p prime, and n ≥ 2 o dd, we have, with notation as ab ove: ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 9 (1) in F t he e quality L p ( n, ω 1 − n p , k ) D 1 / 2 ,p k = q p ( n, k ) E ul p ( n, k ) R n,p ( k ) holds for s ome q p ( n, k ) in Q ∗ ; (2) in fact, q p ( n, k ) = q ( n, k ) ; (3) L p ( n, ω 1 − n p , k ) and R n,p ( k ) ar e non-zer o. As mentioned in the introduction, this, and the corres p o nding pa rts of Con- jecture 3 .18 b elow, ca n b e deduced (with some eﬀo r t) fro m a m uc h more general conjecture of Perrin- Rio u [42, 4.2.2 ]. Remark 2 .18. The conjecture is similar to the result that the r e sidue of ζ p ( s, k ) at s = 1 is related to the Leop oldt reg ulator of O ∗ k through exactly the sa me formula as for the residue of ζ ( s, k ) a nd the Dirichlet regulator [18], with part(4) cor resp onding to the Leopoldt conjecture. How ev er, w e hav e not tr ied to determine if Colmez’s normalizatio n of the sign fo r the regulato r is the sa me as her e, esp ecially given the sign error in the pr o of of Lemma 4.3 in lo c. cit. (see Se c tion 6 ). Remark 2.19. W e c an use a basis of a subgroup of ﬁnite index o f K 2 n − 1 ( k ) / torsio n in the deﬁnition of R n, ∞ ( k ) a nd R n,p ( k ), or even a Q -ba sis of K 2 n − 1 ( k ) Q , without aﬀecting the rationality of q ( n, k ), q p ( n, k ) or their equality . Similarly we can repla ce the Z -basis o f O k with a Q -basis of k in the deﬁnitions o f D 1 / 2 , ∞ k and D 1 / 2 ,p k . Remark 2.20. As is well-known, for k and n as in Conjecture 2.17, by (2.6) ζ k ( s ) at s = 1 − n has a z ero of o rder d and the ﬁrst non-zer o co eﬃcient in its T a ylor expansion, ζ ♯ k (1 − n ), equals (2 π i ) d (1 − n ) (( n − 1)! / 2 ) d D n − 1 / 2 k ζ k ( n ). If we take D 1 / 2 , ∞ k = D 1 / 2 k in (2.16) then w e obtain ζ ♯ k (1 − n ) = (( n − 1)! / 2) d q ( n, k ) D n − 1 k e R n, ∞ ( k ) for Beilinso n’s r enormalized reg ulator e R n, ∞ ( k ) = (2 π i ) d (1 − n ) R n, ∞ ( k ). In computer calculations (( n − 1)! / 2) d q ( n, k ) D n − 1 k often has o nly relatively small prime factor s, so the larg er prime factors in q ( n, k ) corresp ond to D 1 − n k . This pheno menon also o ccurs in the ca lculations for Conjecture 3.18 b elow (see Remar k 7 .8). Remark 2. 21. (1) The thought that L p ( n, ω 1 − n p , k ) is non-zero for n ≥ 2 a nd o dd when k is a totally real Abelian extension of Q is men tioned b y C. Soul´ e in [50, 3.4]. (2) F. Calegari [13] (see also [8]) proved that, for p = 2 and 3, ζ p (3), whic h in those cas es equals L p (3 , ω − 2 p , Q ), is irrational. (More res ults along these lines are describ ed in Remark 3.20(3).) (3) P arts (1) and (2) of Co njecture 2 .17 ho ld when k is a totally real Abelian nu mber ﬁeld, and in fact the corresp onding parts of a m uch stronger conjecture that we shall describ e in Section 3 hold for cy c lotomic ﬁelds (see Pr op osition 4.1 7). (4) W e numerically v eriﬁed part (3) of Conjecture 2.1 7, and its more reﬁned version Conjecture 3.18(4) b elow, in certain case s ; see Remar k 4 .19 a nd Section 7. 3. A motivic version of the co n jecture If E is an y extension of Q , and k / Q is ﬁnite and Galois with Galo is gro up G , then w e let M E = E ⊗ Q k and K 2 n − 1 ( k ) E = E ⊗ Q K 2 n − 1 ( k ) Q , whic h are E [ G ]- mo dules. The g oal of this sectio n is to r e ﬁne Co njectur e 2 .17 to a conjecture for 10 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT Artin mo tives with co eﬃcients in E , or equiv alently , idempo tent s in the g roup r ing E [ G ], when E is a num be r ﬁeld. Deﬁnition 3.1. F or an idemp otent π in E [ G ] we let M E π = π M E and K 2 n − 1 ( M E π ) = π K 2 n − 1 ( k ) E . Now ﬁx a n embedding φ ∞ : k → C . The pairing G × k → C mapping ( σ , a ) to φ ∞ ( σ ( a )) le ads to a n E - bilinear pairing (3.2) E [ G ] × M E → E ⊗ Q C ( eσ , e ′ ⊗ a ) 7→ ee ′ ⊗ φ ∞ ( σ ( a )) . and we co nsider its restric tion ( · , · ) ∞ : E [ G ] π × M E π → E ⊗ Q C . Similarly , r eplacing φ ∞ with a ﬁxed embedding φ p : k → Q p we o btain ( · , · ) p : E [ G ] π × M E π → E ⊗ Q F . where F ⊂ Q p is the topo lo gical clo sure o f φ p ( k ), which is indep endent o f φ p since k / Q is Galois. Lemma 3.3 . L et π in E [ G ] b e an idemp otent . Then dim E ( M E π ) = dim E ( E [ G ] π ) . Pr o of. Since π 2 = π , (3 .2) is identically 0 o n E [ G ] π × M E 1 − π and E [ G ](1 − π ) × M E π . But the determinant of this pair ing is , up to multiplication by E ∗ , equal to D 1 / 2 , ∞ k , hence no n-zero. This implies the le mma.  W e now in tro duce pairings similar to ( · , · ) ∞ and ( · , · ) p but replacing M E π with K 2 n − 1 ( M E π ). If we denote the map K 2 n − 1 ( k ) Q → K 2 n − 1 ( C ) Q induced by φ ∞ by φ ∞∗ and let Φ ∞ be the comp ositio n with the B eilinson regulator map reg ∞ for C , Φ ∞ = reg ∞ ◦ φ ∞∗ : K 2 n − 1 ( k ) Q → K 2 n − 1 ( C ) Q → R ( n − 1 ) ⊂ C , then the pa ir ing G × K 2 n − 1 ( k ) Q → C given by mapping ( σ , α ) to Φ ∞ ( σ ( α )) gives rise to an E -bilinear pairing (3.4) E [ G ] × K 2 n − 1 ( M E ) → E ⊗ Q C ( eσ , e ′ ⊗ α ) 7→ ee ′ ⊗ Φ ∞ ( σ ( α )) and we co nsider its restric tion [ · , · ] ∞ : E [ G ] π × K 2 n − 1 ( M E π ) → E ⊗ Q C . By Lemma 2.15 and the deﬁnition of F the sy n tomic regula tor gives us reg syn : K 2 n − 1 ( O F ) Q → F . If we write φ p ∗ for the compo sition K 2 n − 1 ( k ) Q ∼ = K 2 n − 1 ( O k ) Q → K 2 n − 1 ( O F ) Q , with the second map induced by φ p : k → F , a nd let Φ p be the comp osition reg syn ◦ φ p ∗ : K 2 n − 1 ( k ) Q → K 2 n − 1 ( F ) Q → F , then we can similarly obtain a pairing [ · , · ] p : E [ G ] π × K 2 n − 1 ( M E π ) → E ⊗ Q F . W e now ﬁx ordered E -bases of E [ G ] π , M E π , and K 2 n − 1 ( M E π ). Deﬁnition 3.5. F or ∗ = ∞ or p we let D ( M E π ) 1 / 2 , ∗ be the determinant of the pairing ( · , · ) ∗ , computed with res pec t to our ﬁxed ba ses of E [ G ] π and M E π . ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 11 Note that D ( M E π ) 1 / 2 , ∞ is non-zero b y the pro of of Lemma 3.3, and the same argument works for D ( M E π ) 1 / 2 ,p . Deﬁnition 3.6. If dim E ( E [ G ] π ) equa ls dim( K 2 n − 1 ( M E π )) then for ∗ = ∞ o r p we let R n, ∞ ( M E π ) b e the determinant of the pairing [ · , · ] ∗ , co mputed with respect to our ﬁxed bases of E [ G ] π and K 2 n − 1 ( M E π ). F o r future use we prove the following. Lemma 3.7 . If dim E ( E [ G ] π ) = dim( K 2 n − 1 ( M E π )) then (1) R n, ∞ ( M E π ) /D ( M E π ) 1 / 2 , ∞ is indep endent of the b asis of E [ G ] π , of φ ∞ , and lies in E ⊗ R ; (2) R n,p ( M E π ) /D ( M E π ) 1 / 2 ,p is indep endent of the b asis of E [ G ] π , o f φ p , and lies in E ⊗ Q p . Pr o of. W e prov e the seco nd sta temen t, the pro of of the ﬁrst being entirely simila r . Cho osing a diﬀerent E -ba sis o f E [ G ] π corresp onds to letting an E -linear tra nsfor- mation act o n E [ G ] π , and in the given quo tient the r e s ulting deter minant ca ncels. Since k/ Q is Galois we get a ll possible embeddings of k into Q p by replacing φ p with φ p ◦ σ for σ in G . F or b oth R n,p ( M E ) and D 1 / 2 ,p ( M E ) this corresp onds to letting σ a ct on E [ G ] π , and the resulting deter mina n t cance ls as b efore. That the quotient lies in E ⊗ Q p follows b ecause it lies in E ⊗ F and w e ha ve just prov ed that it is inv ar iant under Ga l( F / Q p ) by Lemma 2 .15.  W e now inv estigate w hen the t wo dimensions in Deﬁnition 3.6 a re equa l. The answer is g iven b y Pro po sition 3.1 2 b elow, but we need some pr eliminary results. Prop ositio n 3.8. If k/ Q is a ﬁ nite Galois extension with Galois gr oup G and π is an idemp otent in E [ G ] then dim E ( K 2 n − 1 ( M E π )) ≤ dim E ( E [ G ] π ) . Equality holds pr e cisely wh en any ( h enc e ev ery ) τ in the image in G of the c onjugacy class of c omplex c onjugation in Gal( Q / Q ) acts by ( − 1 ) n − 1 on E [ G ] π . Pr o of. F or the s tatement we can ﬁrst replace E b y a ﬁnitely generated subﬁeld since π con tains only ﬁnitely man y elements of E , next em bed E into C and use this embedding to enlarge E to C . So we may assume that E = C . According to Theorem 2.2 the pa ir ing (3.4) gives an injection (3.9) R ⊗ Q K 2 n − 1 ( k ) Q → R [ G ] ∨ = Hom R ( R [ G ] , R ) α 7→ f α with f α ( σ ) = 1 (2 π i ) n − 1 Φ ∞ ( σ ( α )) and b y extending the co eﬃcients w e get an injection C ⊗ Q K 2 n − 1 ( k ) Q → C [ G ] ∨ . The image of π ( K 2 n − 1 ( k ) Q ⊗ Q C ) under the last map v anis hes on C [ G ](1 − π ) since (1 − π ) π = 0 so that π K 2 n − 1 ( k ) Q ⊗ Q C injects in to ( C [ G ] / C [ G ](1 − π )) ∨ ∼ = ( C [ G ] π ) ∨ , which pr ov es our ﬁrs t inequality . As for equality , we know by Theo rem 2 .2 that (3.9) has as its image the subspace of R [ G ] ∨ where, under the action o f Gal( Q / Q ) on R [ G ] ∨ via ( ˜ σ f )( σ ) = f ( ˜ σ − 1 σ ), the conjucag y c lass of c o mplex co njugation in Gal( Q / Q ) a cts as multiplication by ( − 1) n − 1 . The same will there fo re hold w ith complex co eﬃcients. Since any ele men t τ in the co njugacy clas s of complex conjugation has order 1 or 2, C [ G ] π deco mpo ses int o eigenspa ces fo r the eigenv alues ± 1 and the desired equality can only hold if τ acts as mult iplication by ( − 1) n − 1 on all of C [ G ] π .  12 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT W e now determine precisely when the equality o f dimensions as in Pro p osition 3.8 can o ccur, and for this we need a preliminar y r esult. Prop ositio n 3.10. L et ψ b e a r epr esentation o f Gal( Q / Q ) over Q that factorizes thr ough the Galois gr oup of a ﬁ nite Galois ex tension of Q and for which ψ ( τ ) acts as multiplic ation by ( − 1) n − 1 for any τ in t he c onjugacy class of c omplex c onjugation in Gal( Q / Q ) . Then the ﬁ x e d ﬁeld of Ker( ψ ) is a ﬁnite Gal ois ex tension of Q that is total ly r e al if n is o dd, and CM if n is even. Pr o of. That the ﬁxed ﬁeld k of Ker( ψ ) is a ﬁnite Galois extension of Q is clear. When n is o dd ψ ( τ ) = 1 for any τ in the co njugacy class of complex co njugation so that k is totally real. F or even n we let ω b e the compo sition Gal( Q / Q ) → Gal( Q ( µ 4 ) / Q ) ∼ → {± 1 } ⊂ Q ∗ . Then Ker( ψ ω ) contains τ a nd Ke r( ψ ) ∩ Ker( ω ), hence its ﬁxed ﬁeld is a totally real Galois extension k + of Q , co nt ained in k ( µ 4 ). Similarly k is contained in the CM ﬁeld k + ( µ 4 ) so, since k is not totally real, it m ust b e CM.  Remark 3. 11. F o r a CM ﬁeld k with k / Q Ga lois, its maximal totally r eal subﬁeld is Galois o ver Q and is the ﬁxed ﬁeld of an elemen t of o rder t w o in the cent re of Gal( k/ Q ), which we shall refer to a s the c o mplex c o njugation of k . Prop ositio n 3.12. L et k / Q b e a ﬁnite Galois extension with Galoi s gr oup G , E any extension of Q , and π an idemp otent of E [ G ] . L et k ′ b e the ﬁxe d ﬁeld of t he kernel of the r epr esentation of G on E [ G ] π . Then for n ≥ 2 the e qu ality dim E ( E [ G ] π ) = dim E ( K 2 n − 1 ( M E π )) holds pr e cisely in the fol lowing c ases: (1) k ′ is total ly r e al and n is o dd; (2) k ′ is a CM ﬁeld, n is even, and the c omplex c onjugation of k ′ acts on E [ G ] π as multiplic ation by − 1 . Pr o of. F rom Pro po sitions 3.8 and 3.10 we see that there ca nnot b e any o ther cases . Conv ersely , by P rop osition 3.8 equality holds in b oth.  W e now r ecall and int ro duce so me termino logy for later use. Deﬁnition 3.1 3 . Let G b e a ﬁnite g roup and E an y extension of Q . (1) If π is a central idempotent of E [ G ] such tha t E [ G ] π is a minimal (non-zero) 2-sided ideal of E [ G ] then π is a primitive central idemp otent. (2) If π is a primitiv e cen tral idemp otent of E [ G ] s uch that E [ G ] π ∼ = M m ( E ) as E -algebr as for some m then we call a primitive idemp otent corr esp ond- ing to π any element in E [ G ] π ⊆ E [ G ] that maps to a matr ix in M m ( E ) that is conjugate to a matr ix with 1 in the upp er left corner and 0’s els e - where. (Since all automorphisms of M m ( E ) as E -algebr a are inner this is independent of the isomo r phism E [ G ] π ∼ = M m ( E ).) Remark 3.14. (1) If E [ G ] π ∼ = M m ( E ) th en an y idemp otent in E [ G ] π c an b e written as a sum of orthogonal primitiv e idempo tent s cor resp onding to π a s one sees immediately b y diagonalizing the matrix A that π ma ps to, whic h satisﬁes A 2 = A . (2) If G = Gal( k / Q ) and π is a primitiv e central idemp o tent of E [ G ] then the dimensions of E [ G ] ˜ π and ˜ πK 2 n − 1 ( k ) E are equal for some non-zero idemp otent ˜ π in E [ G ] π if and only if the sa me holds for one (hence an y) pr imitive idemp otent corres p o nding to π . Indeed, the dimensions for π do no t c hange if we r eplace it with a conjugate in E [ G ], and they a dd up for sums of o rthogona l idemp otents. ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 13 W e now intro duce L -functions, both classical and p -adic, in the following context. Let E be a ﬁnite e x tension of Q , k a n um b er ﬁeld, and consider a character ψ of a representation of G = Gal( k /k ) on a ﬁnite dimensional E -vector space V that factorizes through Gal( k /k ′ ) for some ﬁnite extensio n k ′ of k . Then for ev ery embedding σ : k → C we hav e the Artin L -function L ( s, σ ( ψ ) , k ), with for ev ery prime P of k the r ecipro cal Eul P ( s, σ ( ψ ) , k ) of the Euler factor cor resp onding to P . Under the natural isomorphis m E ⊗ Q C ∼ → ⊕ σ C σ the L ( s, σ ( ψ ) , k ) corresp ond to a cano nic a l E ⊗ Q C -v a lued L -function that we denote by L ( s, ψ ⊗ id , k ). Similarly , for every prime P o f k we hav e a n E ⊗ Q C -v a lued Eul P ( s, ψ ⊗ id , k ) cor resp onding to the Eul P ( s, σ ( ψ ) , k ). W e now move on to the p -adic L -functions, and a s sume that k is totally real, p a prime num ber, a any in teger and ω p the T eichm¨ uller character Gal( Q /k ) → Q ∗ p (2.14). If τ : E → Q p is an embedding and k τ ( ψ ) ω a p is totally real then from Section 2 w e have the p -adic L -function L p ( s, τ ( ψ ) ω a p , k ), whic h is not identically zero. In this ca se, using the natural is o morphism E ⊗ Q Q p ∼ → ⊕ τ Q p,τ , they give us an E ⊗ Q Q p -v a lued p -adic L -function on Z p or Z p \ { 1 } that we denote by L p ( s, ψ ⊗ ω a p , k ). Lemma 3.1 5 . The values of L p ( s, ψ ⊗ ω a p , k ) ar e in E ⊗ Q Q p . Pr o of. Using B r auer induction for ψ (cf. (2.1 0)) it suﬃces to do prov e this when ψ is 1 -dimensional. But then for each τ : E → Q p the function L p ( s, τ ( ψ ) ω a p , k ) can be describ ed a s in (6.3) with l = − 1, from which the result is clea r.  Remark 3.16 . It follows fro m Remar k 2 .13 that L p ( s, ψ ⊗ ω a p , k ) satisﬁes L p ( m, ψ ⊗ ω a p , k ) = Eul p ( m, ψ ⊗ id , k ) L ( m, ψ ⊗ id , k ) for in tegers m ≤ 0 congruent to 1 − a mo dulo φ ( q ), wher e both sides lie in E = E ⊗ Q Q inside E ⊗ Q Q p and E ⊗ Q C resp ectively , and Eul p ( s, ψ ⊗ id , k ) = Y P | p Eul P ( s, ψ ⊗ id , k ) the pro duct be ing ov er a ll pr imes o f k dividing p . Remark 3.17. If E = Q w e shall identify E ⊗ Q C with C , and w r ite L ( s, ψ , k ), etc., instead of L ( s, ψ ⊗ id , k ), etc. Simila r ly we ident ify E ⊗ Q Q p with Q p and write L p ( s, ψ ω a p , k ) instead of L p ( s, ψ ⊗ ω a p , k ). W e now hav e all the ingredients for the gener alization and reﬁnement of Con- jecture 2.17. Starting with a ﬁnite Galois extension k / Q with Galois gro up G , E a ﬁnite extension of Q , and π an idempotent in E [ G ], w e let ψ π be the nat- ural representation of Gal( Q / Q ) on E [ G ] π and χ π its asso ciated c haracter. If dim E ( E [ G ] π ) = dim E ( π K 2 n − 1 ( k ) E ) for some n ≥ 2 then, for a ny pr ime p and any embedding τ : E → Q p , τ ( ψ π ) ω 1 − n p is tr iv ial on the co njugacy cla s s of complex con- jugation in Gal( Q / Q ) b y Prop ositio n 3.8. In par ticular, Q τ ( ψ π ) ω 1 − n p is tota lly r eal and ther efore L p ( n, χ π ⊗ id , Q ) is not iden tically zero. With F ⊂ Q p the topo logical closure of φ p ( k ) as b efore , using or de r ed bases for E [ G ] π , M E π and K 2 n − 1 ( M E π ), we hav e D ( M E π ) 1 / 2 , ∞ and the r e g ulator R n, ∞ ( M E π ) in E ⊗ C as w ell as D ( M E π ) 1 / 2 ,p and R n,p ( M E π ) in E ⊗ F . 14 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT Conjecture 3.18. With n otation as ab ove, if dim E ( E [ G ] π ) = dim E ( π K 2 n − 1 ( k ) E ) for some n ≥ 2 , then (1) in E ⊗ Q C we have L ( n, χ π ⊗ id , Q ) D ( M E π ) 1 / 2 , ∞ = e ( n, M E π ) R n, ∞ ( M E π ) for some e ( n, M E π ) in ( E ⊗ Q Q ) ∗ ; (2) in E ⊗ Q F we have L p ( n, χ π ⊗ ω 1 − n p , Q ) D ( M E π ) 1 / 2 ,p = e p ( n, M E π ) E ul p ( n, χ π ⊗ id , Q ) R n,p ( M E π ) for some e p ( n, M E π ) in ( E ⊗ Q Q ) ∗ ; (3) in fact, e p ( n, M E π ) = e ( n, M E π ) ; (4) L p ( n, χ π ⊗ ω 1 − n p , Q ) a nd R n,p ( M E π ) ar e units in E ⊗ Q Q p and E ⊗ Q F r esp e ctively. Remark 3. 19. (1 ) One sees as in t he pro of of Lemma 3.7 that the v alidity of each part of the co njecture is indep endent of the chosen base s of E [ G ] π , M E π and K 2 n − 1 ( M E π ). (2) Since E ⊗ Q C ∼ → ⊕ σ E σ , wher e the sum is ov er a ll embeddings σ : E → C , an identit y in E ⊗ Q C is equiv alent to the corresp onding identit ies for all s uch embeddings. The sa me ho lds if we replace C with Q p . In Remar k 3.24 w e shall make explicit how part (1) of this conjecture is e q uiv alent with Beilinso n’s co njecture [2, Co njecture 3.4] for a n Artin motive asso ciated with π . Fir st, w e mak e v arious r emarks ab out its dep endence on E , etc., and on its relation with Conjecture 2.17. Remark 3. 20. (1) An equiv alent for Conjecture 3.18(1) can b e for mulated for any idempo ten t π if k / Q is any ﬁnite Galois extension (see the end of Section 5), in which case it is a conjectur e by Gross (see [39, p. 210 ]). In that case it was proved by Beilins on if the action of G = Gal( k/ Q ) on E [ G ] π is Ab elian (see lo c. cit.). One can deduce that it holds for any π in E [ G ] if this action fac tors through S 3 or D 8 (see P rop osition 5.9). (2) If the a c tion of G on E [ G ] is Ab elian then par ts (1)-(3) of Conjecture 3.18 also ho ld for any π to which the conjecture applies (s e e Remar k 4.18). (3) E xtending and simplifying ea rlier work b y F. Calegar i [13], F. Beukers in [8] prov ed that ζ p (2) is irra tional when p = 2 o r 3. This v alue equa ls L p (2 , χω − 1 p , Q ) with χ the primitiv e character o n ( Z / 4 Z ) ∗ for p = 2, and the primitive character on ( Z / 3 Z ) ∗ for p = 3. Moreov er, if χ is the o dd primitive c haracter on ( Z / 8 Z ) ∗ then he also shows tha t L 2 (2 , χω − 1 2 , Q ) is irra tional. It follows that the co njecture holds in full for π corres po nding to the non-triv ial repr esentation of Gal( k / Q ) and n = 2 whe n ( k , p ) is one of ( Q ( √ − 1) , 2), ( Q ( √ − 2) , 2) and ( Q ( √ − 3) , 3). (4) W e hav e veriﬁed numerically that part (4) of the conjecture holds in cer ta in cases; s ee Rema r k 4.19 a s well as Section 7. Remark 3 .21. (1) If π = π 1 + · · · + π m with π 2 i = π i and π i π j = 0 when i 6 = j , then the conjecture for π is implied b y the conjecture for all π i bec ause D ( ⊕ i M E i ) 1 , 2 / ∞ = Q i D ( M E i ) 1 / 2 , ∞ , etc., as one easily se e s by using bas es. (2) If π is in E [ G ] and E ′ is an extensio n of E , then we ma y view π as an element of E ′ [ G ] as well, and the conjectures for M E i and M E ′ i are equiv alent: w e can use the same bases o ver E ′ as o ver E , so that D ( M E i ) 1 / 2 , ∞ = D ( M E ′ i ) 1 / 2 , ∞ ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 15 in E ⊗ C ⊆ E ′ ⊗ C , and the sa me holds for all the other ingredients (including the L -functions). (3) By comparing bases one sees immediately that if π and π ′ are conjugate under the action o f E [ G ] ∗ then the conjectures for π and for π ′ are eq uiv alent. (4) If, for a primitive central idemp otent π i of E [ G ], E [ G ] π i ∼ = M m ( E ) for some m , and π is a primitive idempotent corresp onding to π i , then the conjecture for π i is implied by the co njecture for π . Indeed, we can deco mpo s e π i int o a sum of o rthogona l primitive idempo ten ts as in Remark 3.14(1), and the truth of the conjecture for π i is implied by its truth for each such primitive idemp otent. But all s uch pr imitive idemp otents are conjugate to π hence pa rt (3 ) ab ov e applies. Remark 3 .22. If k / Q is a Ga lois extensio n with Galois gro up G a nd H a subgr oup of G with ﬁxe d ﬁeld k H , then K m ( k ) H Q = K m ( k H ) Q , a result known as Galois descent. With π H = | H | − 1 P h ∈ H h , an idemp o tent in E [ G ], this implies that π H K m ( k ) = K m ( k H ). Remark 3. 23. (1) Let k / Q be a Galois ex tension with Galois g r oup G . If N is a normal subgroup of G corres po nding to k ′ = k N and π N = | N | − 1 P h ∈ N h , then π 2 N = π N and the natural ma p φ : E [ G ] → E [ G/ N ] ha s kernel E [ G ](1 − π N ) and induces an isomor phism E [ G ] π N → E [ G/ N ]. Indeed, it is clear that π N is central in E [ G ] since N is normal in G and tha t 1 − π N is in the k ernel of φ . Also, since N acts tr iv ially on E [ G ] π N this is a n E [ G/ N ]-mo dule generated by one element, so its dimension ov er E cannot be big g er than | G/ N | . Since φ is obviously surjective our claims follo w. Therefor e in this situatio n for an y idempotent π in E [ G/ N ] there is a canonical idempo tent ˜ π lifting π to E [ G ] π N , and the natura l map E [ G ] ˜ π → E [ G/ N ] π is an isomor phism of E [ G ]- a nd E [ G/ N ]-mo dules. Then the statemen ts o f Conjecture 3 .18 for π in E [ G/ N ] or for ˜ π in E [ G ] are equiv a lent . Namely , ˜ π K 2 n − 1 ( k ) E = ˜ π π N K 2 n − 1 ( k ) E = ˜ π K 2 n − 1 ( k ′ ) E = π K 2 n − 1 ( k ′ ) E inside K 2 n − 1 ( k ) E so that w e can use the same bases for either side. The same ho lds for ˜ π ( E ⊗ k ) and π ( E ⊗ k ′ ) inside E ⊗ k . Mor eov er, E [ G ] ˜ π is the pullback to G of the G/ N -repr esentation E [ G/ N ] π , s o that L ( s, E [ G ] ˜ π , Q ) = L ( s, E [ G/ N ] π , Q ) and similarly fo r the p -adic L -functions. (2) If k is a totally r eal num ber ﬁeld k let ˜ k be its (totally real) Galois clo sure ov er Q . Then k = ˜ k H for some subgroup H of G = Gal( ˜ k / Q ), π = | H | − 1 P h ∈ H h is a n idempotent in Q [ G ], and for n ≥ 2 o dd Conjecture 3.18 for π is eq uiv alent to Conjecture 2 .17 for k . Namely , K 2 n − 1 ( k ) Q = π K 2 n − 1 ( ˜ k ) Q ⊆ K 2 n − 1 ( ˜ k ) Q by Re- mark 3.22 and π ˜ k = k so that we can use the sa me Q -bases in each case. Moreov er, Q [ G ] π ∼ = Ind G H (1 H ) = Q [ G ] ⊗ Q [ H ] 1 H with 1 H the trivia l 1 -dimensional r epresenta- tion of H , as one ea sily sees by mapping β in Q [ G ] π to β ⊗ v and P σ a σ σ ⊗ ( λv ) to λ P σ a σ σ π , { v } being a basis of 1 H . By well-known prop erties o f Artin L -functions [40, P rop. 10 .4(iv)] this implies that ζ k ( s ) = L ( n, Q [ G ] π , Q ) and similarly for the p -adic L -functions. Remark 3 . 24. W e make the relation b etw een Conjecture 3.18(1) and Beilinso n’s conjecture [2, Co njecture 3.4] for (Artin) motives explicit since the relation betw een the tw o elements of E ∗ inv olved also sho ws up very explicitly in our computer calculations, sug gesting that the element for the for m ulation at s = 1 − n is simpler 16 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT than for our formulation at s = n (see Remar k 7.8). W e provide some de ta ils since we c o uld no t ﬁnd a deta ile d enough reference in the liter ature. With notation as in Co njecture 3 .18 a nd (2.6) we have (3.25) L (1 − s , χ ∨ π ⊗ id , Q ) = W ( χ ∨ π ⊗ id) C ( χ ∨ π ⊗ id) s − 1 2 (2(2 π ) − s Γ( s )) χ ∨ (1) × (cos( π s/ 2)) n + (sin( π s/ 2)) n − L ( s, χ π ⊗ id , Q ) , with χ ∨ π the dual character of χ π , W ( χ ∨ π ⊗ id) in E ⊗ C , and C ( χ ∨ π ⊗ id) = C ( χ π ⊗ id) in Q ∗ . If m = dim E ( E [ G ] π ), then n + = m a nd n − = 0 for n o dd, and n − = m and n + = 0 for n even. In either ca se L ( s, χ ∨ π ⊗ id , Q ) has a zero of order m at s = 1 − n . Moreover (3.26) W ( χ ∨ π ⊗ id) C ( χ ∨ π ⊗ id) 1 / 2 = e n i m ( n − 1) D ( M E π ) 1 / 2 , ∞ for some e n in E ∗ = ( E ⊗ Q Q ) ∗ by [2 0, Prop os itio ns 5 .5 and 6 .5] since D 1 / 2 , ∞ can be taken to b e the same for M E π and the asso cia ted determinant representation (cf. [31, p.3 60]). Hence the ﬁrst non-v anishing coe ﬃcient in its T aylor expansio n around s = 1 − n , L ♯ (1 − n, χ ∨ π ⊗ id , Q ), equals δ n e n (2 π i ) m (1 − n ) (( n − 1 )! / 2) m C ( χ ∨ π ⊗ id) n − 1 D ( M E π ) 1 / 2 , ∞ L ( n, χ π ⊗ id , Q ) with δ n = ( − 1) m ( n − 1) / 2 when n is o dd, and δ n = ( − 1) m ( n +2) / 2 when n is even. In particular, Conjecture 3.18(1) is equiv alent with (3.27) L ♯ (1 − n, χ ∨ π ⊗ id , Q ) = δ n e n (( n − 1)! / 2) m C ( χ ∨ π ⊗ id) n − 1 e ( n, M E π ) e R n, ∞ ( M E π ) with the renor ma lized regula tor e R n, ∞ ( M E π ) = (2 π i ) m (1 − n ) R n, ∞ ( M E π ). Let us compare this with Beilinson’s co njecture for a motive as s o ciated with π . W e asso cia te motives cov ar iantly to smo oth pro jective v arie ties ov er Q as in [2, § 2 .4]. The Galo is group G acts on the left on k , hence on the right on Sp ec( k ), and we let M π be the motiv e co r resp onding to π under this action. Then G acts on the left on the cohomo logy theo ries on Sp ec( k ) as well a s its K -theory , and after tensoring with E the cor resp onding g roups fo r M π are the ima ges under π . Th us, the relev ant motivic co ho mology of M π is H 1 M ( M π / Z , Q ( n )) = K 2 n − 1 ( M E π ). W e shall nee d the non-degenera te E -bilinear pair ing E [ G ] × M τ : k → C E → E ( P σ a σ σ , ( b τ ) τ ) 7→ P σ a σ b φ ∞ ◦ σ − 1 . It factors through the left E [ G ]-a c tio n on ⊕ τ E (given b y σ (( b τ ) τ ) = ( b τ ◦ σ − 1 ) τ ), hence is trivial on E [ G ](1 − π ) × π ( ⊕ τ E ) and E [ G ] π × (1 − π )( ⊕ τ E ). W e therefo r e obtain a non-degenera te E -bilinear pair ing (3.28) h · , · i : E [ G ] π × π ( ⊕ τ E ) → E that identiﬁes E [ G ] π and π ( ⊕ τ E ) as dual E [ G ]-mo dules. T ensoring (2.1) with E and applying π we get H 1 D ( M π / R , R ( n )) ∼ = π  M τ : k → C E ⊗ R ( n − 1)  . Note that the left hand side is a subspace of the right hand side by (2.1), but b e cause the regulator map (2.3) tensore d with R is injectiv e, and dim E ( K 2 n − 1 ( M E π )) = dim E ( E [ G ] π ) = dim E ( π ( ⊕ τ E )) b y our as sumption o n π a nd since D ( M E π ) 1 / 2 , ∞ 6 = 0 ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 17 (as was noticed rig ht after Deﬁnition 3.5), e q uality m ust hold. F or the Beilinso n re g - ulator we there fo re hav e to compare the tw o E - structures o n det H 1 D ( M π / R , R ( n )) coming from Betti cohomolo gy , H 0 B ( M π / R , Q ( n − 1)) ∼ = π ( M τ : k → C E ⊗ Q ( n − 1)) ⊆ π  M τ : k → C E ⊗ R ( n − 1)  , and tha t induced by the B eilinson regulator map (2.3), H 1 M ( M π / R , Q ( n )) → H 1 D ( M π / R , R ( n )) = π  M τ : k → C E ⊗ R ( n − 1)  . Cho osing a n E -ba s is of π ( ⊕ τ E ), and m ultiplying it by (2 πi ) n − 1 to obtain an E - basis of π ( ⊕ τ E ⊗ Q ( n − 1)), it is ea sy to see that Beilinson’s regulator R Bei for M π satisﬁes R Bei = (2 π i ) m (1 − n ) det[ · , · ] ∞ / det h · , · i = e R n, ∞ ( M E π ) / det h · , · i where all determinants are computed using the chosen E - ba ses. Finally , we compare L -functions. W e hav e H 0 et (Spec( k ) ⊗ Q , Q l ) ∼ = M τ : k → Q Q l ∼ = Q l ⊗ Q  M τ : k → Q Q  so that H 0 et ( M π , Q l ) ∼ = Q l ⊗ π ( ⊕ τ E ) as Q l ⊗ E [ G ]-mo dules. The Q l plays no r o le, and as in (3.28) w e se e that π ( ⊕ τ : k → Q E ) is dual to E [ G ] π as E [ G ]-mo dule. As the motivic L -function uses the geometric rather than the arithmetic F rob enius (cf. [34, p.26]), we obtain that L ( s, M π ) of [2, § 3] is equa l to L ( s, χ π ⊗ id , Q ). If θ : E [ G ] → E [ G ] is the E -linear in volution obtained by replac ing e a ch σ in G with σ − 1 , then we need to c o nsider M 0 π = M θ ( π ) instead of M π . But the ma p E [ G ] π × E [ G ] θ ( π ) → E mapping ( α, β ) to the co eﬃcient o f the neutral element of G in θ ( β ) α is eas ily s een to identify E [ G ] θ ( π ) and E [ G ] π as dual E [ G ]-mo dules, so that L ( s, M 0 π ) = L ( s, χ ∨ π ⊗ id , Q ). 4. More explicit K -gr oups and regula tors In this s ection we ﬁrs t descr ibe a n inductive pr o cedure that conjectura lly g ives K 2 n − 1 ( k ) Q ( n ≥ 2) fo r any num ber ﬁeld k . It is originally due to Zagier [5 4], but we essentially give a reformulation b y Deligne [21]. W e also describ e results concerning Conjecture 3.18 when the action of G on E [ G ] π is Ab elian. In order to descr ib e Zag ier’s co njecture we need the functions (4.1) Li n ( z ) = X k ≥ 1 z k k n ( n ≥ 0) for z in C with | z | < 1 if n = 0 or 1, a nd | z | ≤ 1 if n ≥ 2. In particular, Li 1 ( z ) is the main branch of − log(1 − z ). Using that d Li n +1 ( z ) = Li n ( z ) d log( z ) they extend to multi-v alued analytic functions on C \ { 0 , 1 } . By simultaneously co ntin uing all Li n along the same pa th one obtains sing le -v a lued functions o n C \ { 0 , 1 } (see [54, § 7 ] or [1 9, Remark 5.2]) by putting (4.2) P n ( z ) = π n − 1  n − 1 X j =0 b j j ! (2 log | z | ) j Li n − j ( z )  ( n ≥ 1) , 18 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT with b j the j -th Ber noulli n umber and π n − 1 the pro jectio n of C = R ( n − 1 ) ⊕ R ( n ) onto R ( n − 1). These functions satisfy P n ( ¯ z ) = P n ( z ) as well as (4.3) P n ( z ) + ( − 1) n P n (1 /z ) = 0 and (4.4) P n ( z m ) = m n − 1 X ζ m =1 P n ( ζ z ) when m ≥ 1 a nd z m 6 = 1. W e can now des crib e the conjecture. Let d = [ k : Q ], and fo r n ≥ 2 let e B n ( k ) b e a fr ee Ab elian group on generator s [ x ] ∼ n with x 6 = 0 , 1 in k . Deﬁne e P n : e B n ( k ) →  R ( n − 1) d  + [ x ] ∼ n 7→ ( P n ( σ ( x ))) σ : k → C , with  R ( n − 1) d  + = { ( a σ ) σ in R ( n − 1) d such that a ¯ σ = a σ } . Then we deﬁne inductively , fo r n ≥ 2, d n : e B n ( k ) →    2 ^ Z k ∗ if n = 2 B n − 1 ( k ) ⊗ Z k ∗ if n > 2 by [ x ] ∼ n 7→ ( (1 − x ) ∧ x if n = 2 [ x ] n − 1 ⊗ x if n > 2 , where [ x ] n − 1 denotes the class of [ x ] ∼ n − 1 in B n − 1 ( k ), which is deﬁned as B n ( k ) = e B n ( k ) / Ker(d n ) ∩ Ker( e P n ) . There ar e some universal r elations, one of which is that [ x ] n + ( − 1 ) n [1 /x ] n = 0, a consequence o f (4.3). Conjecture 4. 5. If n ≥ 2 then (1) ther e is an inje ction Ker(d n ) Ker(d n ) ∩ Ker( e P n ) → K 2 n − 1 ( k ) Q with image a ﬁ n itely gener ate d gr oup of r ank e qual to dim Q ( K 2 n − 1 ( k ) Q ) ; (2) Beili nson ’s r e gulator map is given by e P n : Ker(d n ) Ker(d n ) ∩ Ker( e P n ) / / ( n − 1)! e P n ' ' O O O O O O O O O O O K 2 n − 1 ( k ) Q Q σ : k → C reg ∞ ◦ σ ∗    R ( n − 1) d  + c ommu tes. Remark 4.6. F or n = 2 the corr esp onding results were alrea dy known befor e Zagier made his conjecture ([9, 51]; see also [2 6, § 2 ]). F r om the results in [19, § 5 ] o ne o bta ins the fo llowing. ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 19 Theorem 4.7 . L et k b e a numb er ﬁeld and let n ≥ 2 b e an inte ger. Th en t her e is an inje ction Ψ n : Ker(d n ) Ker(d n ) ∩ Ker( e P n ) → K 2 n − 1 ( k ) Q with ﬁnitely gener ate d image, such that the diagr am in Conje ctur e 4.5(2) c ommutes. Remark 4.8. The part o f Zag ier’s c o njecture that remains op en is the q uestion if the r ank of the imag e o f Ψ n equals dim Q ( K 2 n − 1 ( k ) Q ). F or a cycloto mic ﬁeld k this holds for an y n ≥ 2 as we sha ll r e c all in Example 4.10 b elow, but it is also known for arbitra r y num ber ﬁelds fo r n = 2 , as mentioned ab ov e, or n = 3 (see [25, § 3] and [2 7, Appendix]). Remark 4.9. Although not stated explicitly in [19], it is clear from the construction there that the map Ψ n in Theorem 4.7 is natural in k . In pa rticular, if k / Q is Galois then Gal( k / Q ) a c ts on K er(d n ) / Ker(d n ) ∩ Ker( e P n ) (thro ug h the obvious action on the generato rs [ x ] ∼ n ) as well as on K 2 n − 1 ( k ) Q , and Ψ n is Gal( k / Q )-equiv ariant. Example 4.10 . If k is a cyclotomic ﬁeld then Zagier ’s c onjecture is kno wn in full. By Theorem 4.7 it suﬃces to see that the rank of the imag e of Ψ n equals dim Q K 2 n − 1 ( k ) Q . F or a n N -th ro ot of unity ζ 6 = 1, N [ ζ ] n is in Ker(d n ) / Ker(d n ) ∩ Ker( e P n ). If k = Q ( ζ ) for a pr imitive N -th roo t of unit y ζ with N > 2 then dim Q K 2 n − 1 ( k ) Q = [ k : Q ] / 2 for n ≥ 2, and the N [ ζ j ] n with 1 < j < N / 2 a nd gcd( j, N ) = 1 have R -indep endent images under e P n [54, pp.420– 422] so that they are Z -indep endent in Ker(d n ) / Ker(d n ) ∩ Ker( e P n ), and the same holds fo r their images under the injectiv e map Ψ n . F or k = Q one can easily chec k directly from Theorem 4 .7 a nd (4.2) that for o dd n ≥ 2 the element 2Ψ ([ − 1] n ) is non-zero. F o r the p -adic regulator w e need to desc r ib e the p -adic p olyloga rithms intro duced in [16]. W e ﬁrst make a choice of a bra nch of the p -adic logarithm. Reca ll that a homomorphism log p : O ∗ C p → C p is unique ly determined by the re q uirement that for | x | p < 1 it is g iven by the usual p ow er s eries for log(1 + x ). T o extend it to a homomorphism lo g p : C p ∗ → C p it suﬃces to make a n ar bitrary choice of lo g p ( p ). An y suc h e x tension will b e called a bra nch of the p -adic po lylogar ithm. In what follows we ﬁx one such choice. Coleman ﬁrst pro duced the p -a dic p o ly logarithm out of a more extensive theory of what is now called Co leman integration. F o r the p -adic p olyloga rithm it is, how ever, po ssible to give a more elementary and explicit theory . T o do this, co nsider the class o f functions f : C p \ { 1 } → C p that satisfy the following pro p erties: (1) for a n y a ∈ C p with | a − 1 | p = 1 we hav e a p ower ser ies expans ion for f ( z ) in z − a that conv erges for | z − a | p < 1 ; (2) for 0 < | z − 1 | p < 1 (resp. | z | p > 1) f ( z ) is given by a po lynomial in log p ( z − 1) (res p. log p (1 /z )) with coeﬃcients that are Laurent series in z − 1 (r esp. 1 /z ), conv ergent for 0 < | z − 1 | p < 1 (resp. | z | p > 1). It is easy to see that diﬀerentiation sends this class surjectively on to itself. The p -adic p olylog arithms a re functions Li n,p ( z ) ( n ≥ 0) in this cla ss with the following prop erties: (1) Li 0 ,p ( z ) = z 1 − z ; (2) Li n,p (0) = 0 for n ≥ 0; (3) dLi n +1 ,p ( z ) = Li n,p ( z ) d z z for n ≥ 0; 20 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT (4) for every n ≥ 0 there is a power se r ies g n ( v ), co nv ergent for | v | p < p 1 / ( p − 1) , such that g n (1 / (1 − z )) = Li n,p ( z ) − 1 p n Li n,p ( z p ) when | z − 1 | p > p − 1 / ( p − 1) . Note that 1 /z has a s ingularity at 0 but Li n,p ( z ) / z doe s not, beca use of the as- sumption that Li n,p (0) = 0. In fact, it is easy to see that L i n,p ( z ) is given b y the p -adically conv ergent p ow er se ries (4.1) for | z | p < 1. In [6] tw o of the authors o f the present work describ ed an eﬃcient algo rithm for the computation of p -a dic po lylogar ithms up to a prescrib ed precision. This will b e used in Section 7 to co mpute the p -adic regula tor for n ≥ 2 that w e now pro ceed to des crib e in terms o f suitable combinations of log n − m p ( z )Li m ( z ). By Prop ositio n 6.4 and the corr ect version of P rop osition 6.1 of [16] the Li n,p ( z ) for n ≥ 0 and z in C p \ { 1 } sa tisfy Li n,p ( z ) + ( − 1) n Li n,p (1 /z ) = − 1 n ! log n p ( z ) when z 6 = 0, a nd (4.11) Li n,p ( z m ) = m n − 1 X ζ m =1 Li n,p ( ζ z ) when m ≥ 1 a nd z m 6 = 1. F o r z in C p \ { 0 , 1 } and ﬁx e d n ≥ 2 , we deﬁne (4.12) P n,p ( z ) = n − 1 X j =0 c j log j p ( z )Li n − j ( z ) ( n ≥ 2) for any c j in C p satisfying c 0 = 1 and P n − 1 j =0 c j / ( n − j )! = 0, so that P n,p ( z ) + ( − 1) n P n,p (1 /z ) = 0. This is the case for c j = b j with b j the j -th Berno ulli num b ers as befor e, although the resulting formula is diﬀerent from (4.2). Another p ossible natural ca ndidate for the function P n,p ( z ) is L n ( z ) + L n − 1 ( z ) log p ( z ) / n , wher e L n ( z ) = n − 1 X m =0 ( − 1) m m ! log m p ( z )Li n − m ( z ) (see [5 , (1.3 ) and Remark 1.5]). The relations corr esp onding to (4.3) and (4.11) for P n,p ( z ) ( n ≥ 2) are then P n,p ( z ) + ( − 1) n P n,p (1 /z ) = 0 when z 6 = 0 , 1, and (4.13) P n,p ( z m ) = m n − 1 X ζ m =1 P n,p ( ζ z ) when m ≥ 1 a nd z m 6 = 0 , 1. F o r any br a nch of the p -adic log arithm one then has the fo llowing r esult for the p -adic regula tor map (see [5 , Theorem 1.1 0 or page 9 09]). Since its formulation depe nds on the em b edding of k we deal with one s uch embedding a t a time. Theorem 4 .14. L et k b e a numb er ﬁeld, and let F ⊂ Q p b e the top olo gic al closur e of the Galois closur e of any emb e dding k → Q p . F or σ : k → F let B σ n ( k ) = h [ x ] n | σ ( x ) , 1 − σ ( x ) ar e in O ∗ C p i ⊆ B n ( k ) = e B n ( k ) Ker(d n ) ∩ Ker( e P n ) . ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 21 Then e B n ( k ) → F [ x ] n 7→ P n,p ( σ ( x )) induc es a map P σ n,p : B σ n ( k ) → F and t he solid arr ows in B σ n ( k ) ∩ Ker(d n )   / / ( n − 1)! P σ n,p + + X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Ker(d n ) Ker(d n ) ∩ Ker( e P n ) Ψ n / / ( n − 1)! P σ n,p ' ' K 2 n − 1 ( k ) Q reg p ◦ σ ∗   F form a c ommutative diagr am. Remark 4.15. It was conjectur e d in [5, Conjecture 1.14] that the dotted arrow exists and that the full dia gram co mm utes. This is known to hold for N [ ζ ] n if ζ is any N -th ro o t of unity o ther than 1 (see Theo r em 1 .12 of lo c. cit.). Remark 4.16. The for mulae for the complex a nd p -a dic regula tors involv ed in The- orems 4 .7 and 4.1 4 are stated up to sig n in [19, Prop ositio n 4.1] (with a normalizing factor) and [5, Theore m 1.1 0(2) or Prop ositio n 7.14] resp ectively , since they dep end on the c hoice of Ψ n , which is na tural only up to sign. In the p -adic ca se it is c le ar from [5, P rop osition 7.1 0] that the s ign is ( − 1 ) n if the “relativity isomor phism” in the p -a dic case is norma lized as in Prop o s ition 5.7 of lo c. cit. and the K -theoretic relativity isomor phism as in (3 .1) of loc. cit. cor resp onds to this under the naturality of the reg ulator map. In order to s ee that the sign fo r the co mplex regula tor is the same with the sa me choice of Ψ n , we observe that one can use the techniques of [5, Appendix A and Section 5] to describ e the tar get for the co mplex regula tor o f [ z ] n with z in C \ { 0 , 1 } in [19] b y means of a c o mplex with the same for mal structure as in [5, Deﬁnition 5.2], but using R ( n − 1 )-v a lued C ∞ -forms ψ with loga rithmic p oles on copies o f ( P 1 C \{ 1 , z } ) j indexed b y inc r easing functions f : [1 , . . . , j ] → [1 , . . . , n − 1]. Using the maps ψ 7→ R ( P 1 C ) j d a rg( t f (1) ) ∧ · · · ∧ d a rg( t f ( j ) ) ∧ ψ for a j -form ψ with index f : [1 , . . . , j ] → [1 , . . . , n − 1], and suitable normalizing factor s, one writes down a n R -linear ma p Π ∞ on suc h forms, with v a lues in R ( n − 1), that v a nishes on exact forms and satis ﬁes the a na logous nor malization c o ndition a s Π = Π p in [5, Pr op osition 5.7]. Moreover, as in [19, Section 2.5] one sees that the sub c om- plex of fo r ms on the comp onent where j = n − 1 that v a nis h if some t l = 0 or ∞ gives a q uasi-isomo rphism under inclusion, and that the ima ge of the regulator of [ z ] n is given b y ε n ( z ) as in Section 4 of loc. cit. (up to sign since one has to use the cor rect formula for ω n , whic h can be obta ined as in [5, Section 6]). Using the computatio ns on pages 236 and 23 7] of [19] one then sees that Π ∞ maps ε n ( z ) to ( − 1) n ( n − 1 )! P ♭ n ( z ), where P ♭ n ( z ) = π n − 1  P n − 1 l =0 1 l ! ( − log | z | ) l Li n − l ( z )  , which induces the sa me map o n Ker(d n ) / Ker(d n ) ∩ Ker( e P n ) as ( − 1) n ( n − 1)! P n ( z ). Since the relativit y isomorphism in K -theory only dep ends on a c hoice of t l = 0 or ∞ for l = 1 , . . . , n − 1 this shows that the signs in the formulae for the complex and p -adic reg ula tors o f [ z ] n match for the same choice of Ψ n , a nd we adjust Ψ n so that those signs disapp ear. 22 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT W e conclude this section by consider ing Conjecture 3.18 when the action of G on E [ G ] π is Abelian. Prop ositio n 4.17. L et N ≥ 2 , k = Q ( µ N ) , G = Gal( k / Q ) = ( Z / N Z ) ∗ wher e a in ( Z / N Z ) ∗ c orr esp onds t o ψ a in G satisfying ψ a ( ζ ) = ζ a for al l r o ots of unity ζ in k . Assume that E c ontains a r o ot of unity of or der e qual to the ex p onent of G and let π in E [ G ] b e the idemp otent c orr esp onding to an irr e ducible char acter χ of G . Then p arts (1), (2) and (3) of Conje ct u r e 3.18 hold for π and n ≥ 2 if χ ( − 1) = ( − 1) n − 1 . Pr o of. That Conjecture 3.18(1) holds in this cas e is well-known (cf. [39]) but w e make it ex plic it here for pa rt (3) of the conjecture (cf. [30, Pro p o sition 3.1 ]). W e deal with the case N = 2 later s o by Remark 3.23(1) we may assume N > 2 and χ pr imitiv e. The idemp otent π cor resp onding to χ is | G | − 1 P ′ N a =1 χ − 1 ( a ) ⊗ ψ a , where the prime indicates that we sum ov er a that ar e relatively prime to N . Note tha t by our assumption on χ ( − 1) and Prop os ition 3.8 the dimension of K 2 n − 1 ( M E π ) equals 1. Fixing a gener ator ζ of µ N ⊂ k ∗ we use π , π (1 ⊗ ζ ) and π (1 ⊗ [ ζ ] n ) as bas is vectors o f E [ G ] π , M E π and K 2 n − 1 ( M E π ) resp ectively , where we simplify notation by writing [ ζ ] n instead of Ψ n ([ ζ ] n ) as in Theo rem 4.7. In order to verify (1) in E ⊗ Q C it suﬃces b y Remar k 3.19(2) to consider all embeddings σ : E → C . If η = φ ∞ ( ζ ) with φ ∞ as in Section 3 then th e σ - comp onent of D ( M E π ) 1 / 2 , ∞ bec omes D ( M E π ) 1 / 2 , ∞ σ = | G | − 1 N X a =1 ′ χ − 1 σ ( a ) η a = N | G | − 1 N X a =1 ′ χ σ ( a ) η − a ! − 1 by [3 6, (3) o n p.84]. Similarly , by Theorem 4 .7 the σ -comp onent of R n, ∞ ( M E π ) is R n, ∞ ( M E π ) σ = ( n − 1)! | G | − 1 N X a =1 ′ χ − 1 σ ( a ) P n ( η a ) = ( n − 1)! | G | − 1 N X a =1 ′ χ − 1 σ ( a )Li n ( η a ) bec ause P n ( η a ) = (Li n ( η a ) + χ − 1 σ ( − 1)Li n ( η − a )) / 2 by our assumption on χ ( − 1), where Li n is computed using the pow er series in (4.1). They r elate to the σ - comp onent of L ( n, χ, Q ) via L ( n, χ σ , Q ) = ( n − 1)! − 1 R n, ∞ ( M E π ) σ /D ( M E π ) 1 / 2 , ∞ σ according to (2) on pa g e 1 72 o f [16] with the cor rect sig n in the exponent o f the Gauss sum as used here (cf. [5 4, p. 421 ]). Since L ( n, χ σ , Q ) 6 = 0 this a lso shows that π (1 ⊗ [ ζ ] n ) gives a ba sis of K 2 n − 1 ( M E π ) as cla imed. Similarly , for (2) and (3) we consider all embedding s τ : E → Q p . If φ p ( ζ ) = η p then th e τ -comp onent of D ( M E π ) 1 / 2 ,p is N | G | − 1 ( P ′ N a =1 χ τ ( a ) η − a p ) − 1 and the τ - comp onent of R n,p ( M E π ) equals ( n − 1)! | G | − 1 P ′ N a =1 χ − 1 τ ( a ) P n,p ( η a p ) b y Re ma rk 4.1 5, independent of our assumption on χ ( − 1 ). By (3) on page 172 o f [16] we have L p ( n, χ τ ω 1 − n p , Q ) = ( n − 1)! − 1 Eul p ( k , χ τ , Q ) R n,p ( M E π ) τ /D ( M E π ) 1 / 2 ,p τ . Therefore Conjecture 3.18(1)-(3) hold with e ( n, M E π ) = e p ( n, M E π ) in Q ∗ ⊆ E ∗ . Now a ssume that N = 2 so that χ is the trivial character 1, π = 1 and n ≥ 2 is o dd. T aking 1, 1 ⊗ 1 a nd 1 ⊗ [ − 1] n as bases of E [ G ] = E , M E π = E ⊗ Q and K 2 n − 1 ( Q ) E we ﬁnd from Theor e m 4.7, (4.1), (4.2) a nd (4.4) that L ( n, 1 , Q ) = ζ ( n ) = 2 n − 1 1 − 2 n − 1 P n ( − 1) = 2 n − 1 ( n − 1)!(1 − 2 n − 1 ) R n, ∞ ( M E 1 ) , ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 23 again pr oving that [ − 1] n gives a basis for K 2 n − 1 ( Q ) Q . Coleman pr ov es that L p ( n, ω 1 − n p , Q ) = (1 − p − n ) lim x → 1 ′ Li n,p ( x ) where the limit is taken in any subﬁeld of C p that is of ﬁnite ramiﬁca tion degr e e ov er Q p . But by (4.1 2) a nd (4.1 3) we ca n rew r ite this as L p ( n, ω 1 − n p , Q ) = (1 − p − n )2 n − 1 1 − 2 n − 1 P n,p ( − 1) = (1 − p − n )2 n − 1 ( n − 1)!(1 − 2 n − 1 ) R p n, ∞ ( M 1 ) .  Remark 4.18 . The π in the pr o of of P rop osition 4.17 a re primitive in the sense of Deﬁnition 3.13 . So, by Remarks 3.21(4) and 3.23(1), if Conjecture 3 .18 applies to π in E [ G ], then pa r ts (1 ), (2) a nd (3) o f it hold for π for a ny E if the a ction o n E [ G ] π is Ab elian. In particular, parts (1) and (2) of Conjecture 2.17 hold for a ny totally real Ab elian num ber ﬁeld by Remar k 3.21. Remark 4. 19. Computer calculations show that L p ( n, ω 1 − n p , Q ) is in Q ∗ p when p = 2 , . . . , 19 and n = 2 , . . . , 20 is o dd, verifying Conjecture 2 .17(3) in those c a ses. Similarly , with notation a nd as sumptions as in Pro p o sition 4.1 7, for the 47 0 primi- tive characters χ of Gal( Q ( µ N ) / Q ) = ( Z / N Z ) ∗ with 2 ≤ N ≤ 5 0, L p ( n, χ ⊗ ω 1 − n p , Q ) lies in ( E ⊗ Q p ) ∗ for those v alues of p and n = 2 , . . . , 20 whenever χ ( − 1) = ( − 1) n − 1 . Thu s Conjecture 3.18(4) a lso holds in tho se ca ses. 5. Computing K -gr oups in practice Let n ≥ 2. It follows from Theorem 4 .7 that Za gier’s appr oach as described in Sectio n 4 can be used to obtain at least some part of K 2 n − 1 ( k ) Q . Using the notation in tro duced in Section 4, in order to carry this approach out in practice one star ts with tw o sets S ⊆ S ′ of primes o f O k and co nsiders only [ x ] ∼ n where x is an S - unit and 1 − x is an S ′ -unit. When we consider several h undred o f such [ x ] ∼ n ’s (and avoid using b oth [ x ] ∼ n and [1 /x ] ∼ n since [ x ] n + ( − 1 ) n [1 /x ] n = 0 a s mentioned in Section 4) this metho d is well suited to computer calculatio ns. The only p oint in Za gier’s appr oach that cannot b e c a rried out a lgebraica lly is to deter mine which elements in Ker( ˜ d n ) are actually in Ker( ˜ d n ) ∩ Ker( e P n ). This is do ne us ing standa rd metho ds for ﬁnding linear rela tions b etw een the e P n ( α j )’s for a ba s is { α j } of Ker( ˜ d n ) with sma ll integral co eﬃcients. T ypically , we used ab out the ﬁrst 50 de c ima ls after the decimal p oint o f a ll the poly lo garithms o f all elements of k inv olved in the α j embedded into C in a ll p ossible wa ys. The relations that were found this way w ere then veriﬁed to hold up to at lea s t 30 additional decimal plac e s . As a ﬁna l chec k, when Ke r(d n ) had the same rank as K 2 n − 1 ( k ) the n umber q in (2.4) (but now using the regulato r e V n ( k ) for the subgroup of K 2 n − 1 ( k ) Q we obtained ra ther than V n ( k ); cf. Remar k 2.19) was computed with a working precision o f up to 12 0 decimals. It turned o ut that its recipr o cal lo o ked ratio nal rather co nvincingly . F o r example, for the cubic ﬁeld k = Q [ x ] / ( x 3 − x 2 − 3 x + 1) with discrimina nt 148, o ne o f the ﬁelds used in Ex a mple 7.1, we ﬁnd for ζ k ( n ) | D k | 1 / 2 / e V n ( k ) the v alues − 1 . 75 31044 5580 71585098612125639152666 179693206722 × 10 − 2 ( n = 3) − 4 . 11 70685 8840 62518549452525064367732 455835167754 × 10 − 9 ( n = 5) 24 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT with recipro ca ls − 57 . 0 41666 6666 6 6666666666666666666666666666666662 ( n = 3) − 242 89126 5 . 599 9 9999999999999999999999999999999998 ( n = 5) . As the calculation of a bas is of K 2 n − 1 ( k ) Q , using the method outlined abov e, tends to take (substantially) lo ng er when [ k : Q ], the absolute v alue of the discrim- inant o f k , or n increase, we des crib e tw o straightforward metho ds to reduce this. They , and mo st o f our arg ument s b elow, will rely on Remar k 3 .22. Metho d 5.1. If k is an Ab elian extension of Q , henc e is c ontaine d in a cyclotomic ﬁeld k ′ , then we c an ﬁnd K 2 n − 1 ( k ) Q by c omputing K 2 n − 1 ( k ′ ) Q as in Example 4.10 and applying Rema rk 3.22 ( using Remark 4.9 ) . Since parts (1), (2) and (3) of Conjecture 3.1 8 are k nown fo r a ll idemp otents of E [ G ] to which it applies if G is Ab elian by Remark 3 .21 and P rop osition 4.17, this will not b e used fo r verifying Co njecture 3.18. Instead, in the cases tha t w e shall consider we r ely o n the following metho d in or de r to ﬁnd the K -g roups. Metho d 5.2. L et k / Q b e Galois with Galois gr oup G , E / Q any ex tension, and M an irr e ducible E [ G ] -mo dule in K 2 n − 1 ( k ) E . If H i s a sub gr oup of G with M H 6 = 0 then M ⊆ E [ G ] · K 2 n − 1 ( k H ) Q . If for every irr e ducible M we c an take H 6 = { e } , then we r e duc e to ﬁnding K 2 n − 1 ( l ) Q to gether with the action of G on it inside K 2 n − 1 ( k ) Q for pr op er subﬁelds l of k . W e now dis cuss in Examples 5.3, 5.5 and 5.7 how to use Metho d 5.2 for certain Galois extensions k / Q . In the decompo sition of K 2 n − 1 ( k ) E according to central idempo ten ts π i of E [ G ] as in Section 3 we concentrate o n those primitive central π i for which the a ction on E [ G ] π i is not Ab elian a nd describ e cor resp onding primitive idempo ten ts π (a s in Deﬁnition 3 .13(2)) for later use. Due to res trictions on when we ca n calc ula te the p -adic L - functions numerically (see Section 6) we only consider k that a re Abelia n over a qua dr atic subﬁeld. Of cour se Prop ositio n 3.1 2 describ es when Co njecture 3.18 a pplies to π , but we work out the structure o f the K -groups and the Galois action in more detail. Example 5.3. Let k / Q be an S 3 -extension with quadratic sub extensio n q . With E = Q and σ in G of order three, π 1 = ( e + σ + σ 2 ) / 3 and π 2 = 1 − π 1 are orthogo nal c ent ral idempotents with π 2 primitive (as in Deﬁnition 3.13(2)). In Q [ G ] = Q [ G ] π 1 ⊕ Q [ G ] π 2 the ﬁrst summand consists o f Abelian representations of G and the last of t wo copies o f the irreducible 2 -dimensional represent ation V of G . A cor resp onding primitive idemp otent for π 2 is π = π 2 π H for an y subgroup H = h τ i of G o f order 2 where π H = ( e + τ ) / 2 . By Remark 3.22 we hav e π 1 K 2 n − 1 ( k ) Q = K 2 n − 1 ( q ) Q so that (5.4) K 2 n − 1 ( k ) Q = K 2 n − 1 ( q ) Q ⊕ π 2 K 2 n − 1 ( k ) Q . The last summand is a Q [ G ]-mo dule isomor phic to V t n for some t n ≥ 0. As V = π V ⊕ τ ′ π V for any τ ′ 6 = τ of order 2 , we ﬁnd, if c = k H , π 2 K 2 n − 1 ( k ) Q = π 2 K 2 n − 1 ( c ) Q ⊕ τ ′ π 2 K 2 n − 1 ( c ) Q again b y Remark 3.2 2. So w e reduce to the calcula tio n of K 2 n − 1 ( q ) Q and K 2 n − 1 ( c ) Q together with the action o f G on the latter. A dimension co unt in (5 .4) using Theorem 2.2 determines t n . W e distinguis h t wo cases. ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 25 Case 1. If q is r eal then t n = 0 for n even and 2 for n o dd. Case 2. If q is ima ginary then t n = 1. Conjecture 3.18 applies to π when t n = 2 . Example 5.5. Let k / Q b e a D 8 -extension wher e D 8 = h σ , τ | σ 4 = τ 2 = ( σ τ ) 2 = e i . W e ﬁx an isomorphism G = Gal( k / Q ) ∼ = D 8 and use notation as in Figur e 5 .1. { e } o o o o o o o o o o o       ? ? ? ? ? ? O O O O O O O O O O h τ i h σ 2 τ i h σ 2 i h σ τ i h σ 3 τ i h τ , σ 2 i ? ? ? ? ? ?      h σ i h σ τ , σ 2 i ? ? ? ? ?      D 8 ? ? ? ? ? ?       k o o o o o o o o o o o o o       ? ? ? ? ? ? O O O O O O O O O O O O O k 1 k 2 k 3 k 4 k 5 q 1 ? ? ? ? ? ?       q 2 q 3 ? ? ? ? ? ?       Q ? ? ? ? ? ?       Figure 5.1. The subg r oups of G = D 8 and their ﬁxed ﬁelds in k / Q . Then Q [ G ] = Q [ G ] π 1 ⊕ Q [ G ] π 2 where π 1 = (1 + σ 2 ) / 2 and π 2 = 1 − π 1 are orthogona l central idemp otents. The ﬁr s t term is an Abe lia n re presentation of G and t he second as a Q [ G ]-module is isomorphic to V 2 for V the irreducible 2-dimensional representation of G ov er Q . Then π 2 is a primitive central idemp otent a nd π = π 2 π H a c orresp onding pr imitive idemp otent if we let H = h τ i and π H = (1 + τ ) / 2. Since π 1 K 2 n − 1 ( k ) Q = K 2 n − 1 ( k 3 ) Q by Rema rk 3.22, for K 2 n − 1 ( k ) Q we ﬁnd (5.6) K 2 n − 1 ( k ) Q = K 2 n − 1 ( k 3 ) Q ⊕ π 2 K 2 n − 1 ( k ) Q with the second term a Q [ G ]-mo dule isomorphic with V t n for so me t n ≥ 0 . If τ ′ in G has order tw o a nd do es no t co mm ute with τ then V = π V ⊕ τ ′ π V , so b y Remark 3 .22 we obtain K 2 n − 1 ( k ) Q = K 2 n − 1 ( k 3 ) Q ⊕ π 2 K 2 n − 1 ( k 1 ) Q ⊕ τ ′ π 2 K 2 n − 1 ( k 1 ) Q and we reduce to the calculation of K 2 n − 1 ( k 3 ) Q and K 2 n − 1 ( k 1 ) Q together with the action of G on the la tter. W e can ﬁnd t n by counting dimensions in (5.6), and distinguish thre e cases. Case 1. F or k 3 and k totally real t n = 0 for n even and t n = 2 for n o dd. Case 2. F or k 3 totally real but k not t n = 2 for n even and t n = 0 for n o dd. Case 3. F or k 3 not tota lly real t n = 1 . Again Conjecture 3.18 applies to π when t n = 2. Example 5.7. Let k / Q b e an S 3 × Z / 3 Z -ex tension, iden tify G = Gal( k / Q ) and S 3 × Z / 3 Z via a ﬁxed isomorphism, a nd choos e a generator σ o f A 3 ⊂ S 3 . Let q be the quadratic subﬁeld of k , s ′ the ﬁxed ﬁeld of A 3 × { 0 } , s ′′ the ﬁxed ﬁeld of { e } × Z / 3 Z and s i for i = 1, 2 the ﬁxed ﬁe ld of H i = h ( σ , i ) i . If E is an extensio n of Q tha t contains a pr imitive third ro o t of unity ζ 3 then (5.8) E [ G ] = E [ G ] π 1 ⊕ E [ G ] π 2 ⊕ E [ G ] π 3 ⊕ E [ G ] π 4 ∼ = E [ Z / 6 Z ] ⊕ W 2 1 ⊕ W 2 ζ 3 ⊕ W 2 ζ 2 3 26 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT where π 1 = (( e, 0) + ( σ, 0) + ( σ 2 , 0 )) / 3 π 2 = (1 − π 1 )(( e, 0) + ( e, 1) + ( e, 2)) / 3 π 3 = (1 − π 1 )(( e, 0) + ζ 2 3 ( e, 1 ) + ζ 3 ( e, 2 )) / 3 π 4 = (1 − π 1 )(( e, 0) + ζ 3 ( e, 1 ) + ζ 2 3 ( e, 2 )) / 3 are o rthogonal central idempotents of E [ G ] s atisfying π 1 + π 2 + π 3 + π 4 = ( e , 0 ), and W a is the irre ducible represe ntation o f G given by ( ρ, m )( v ) = a m ρ ( v ) for v in the irreducible 2-dimensio nal represe n tation of S 3 . With π H i = 1 3 P h ∈ H i h , a primitive idempo ten t corresp onding t o the primitiv e central idemp otent π j ( j = 3, 4) is π = π j π H i . Using Remark 3.22 for A 3 × { 0 } and { e } × Z / 3 Z we see tha t K 2 n − 1 ( k ) E = π 1 K 2 n − 1 ( k ) E ⊕ π 2 K 2 n − 1 ( k ) E ⊕ π 3 K 2 n − 1 ( k ) E ⊕ π 4 K 2 n − 1 ( k ) E = K 2 n − 1 ( s ′ ) E ⊕ (1 − π 1 ) K 2 n − 1 ( s ′′ ) E ⊕ π 3 K 2 n − 1 ( k ) E ⊕ π 4 K 2 n − 1 ( k ) E . Because W a = W H 1 a ⊕ W H 2 a when a 6 = 1 we hav e π j K 2 n − 1 ( k ) E = π j K 2 n − 1 ( s 1 ) E ⊕ π j K 2 n − 1 ( s 2 ) E ( j = 3 , 4) so that we reduce to the calcula tion o f K 2 n − 1 ( s ) Q for the s extic s ubﬁelds s o f k , together with the action o f G when s = s ′′ , s 1 or s 2 . W e hav e π 3 K 2 n − 1 ( k ) E ∼ = W t n ζ 3 and π 4 K 2 n − 1 ( k ) E ∼ = W t n ζ 2 3 for the same t n as one sees b y ta k ing E = Q ( ζ 3 ) and consider ing the obvious actio n o f Gal( E / Q ) on E [ G ]. Since the dimension o f (1 − π 1 ) K 2 n − 1 ( s ′′ ) E was obtained in E xample 5.3 we g et the following r esults for t n . Case 1. F or q re a l t n = 0 when n is even and t n = 2 when n is o dd. Case 2. F or q imag inary t n = 1 . Conjecture 3.18 applies to π cor resp onding to π j ( j = 3 , 4) when t n = 2 . W e ﬁnish this sec tio n with a r esult that was pr omised in Remark 3.20(1). Note that the assumption dim E ( E [ G ] π ) = dim E ( π K 2 n − 1 ( k ) E ) is not needed here: we consider the reg ulator map with v alues in π ( ⊕ τ : k → C E ⊗ R ( n − 1)) + obtained from (2.3), and use Beilinso n’s reg ulator R Bei comparing E -s tructures as in Re- mark 3.24. Beilins on’s c onjecture [2, Conjecture 3.4] states that the order of v an- ishing at s = 1 − n of L ( s, χ ∨ π ⊗ id , Q ) should b e dim E ( π K 2 n − 1 ( k ) E ) and tha t the ﬁrst non-v anishing co e ﬃcient at s = 1 − n equals L ♯ (1 − n, χ ∨ π ⊗ id , Q ) = e R Bei for some e in E ∗ . Prop ositio n 5. 9. L et k b e a G = S 3 -extension ( r esp. G = D 8 -extension ) of Q , and let n ≥ 2 . Then this c onje ctu r e holds for any primitive idemp otent π c orr esp onding to π 2 in Example 5.3 ( r esp. Example 5.5 ) . Pr o of. Clear ly it suﬃces to prov e this when E = Q . If G = S 3 then b y Re- mark 3 .21(3) we may as sume π H = π 1 π H + π with π 1 π H and π = π 2 π H orthogo nal idempo ten ts, where w e use notation as in Exa mple 5.3. T he co njectur e for π H follows fro m Theorem 2.2 for c = k H by the ar guments of Remark 3.23 and the functional eq ua tion (2.6) of ζ c ( s ). Similarly , it holds for π 1 π H = 1 6 P g ∈ G g , where it corresp onds to that theo rem fo r k G = Q , and it follows tha t it m ust hold for π as well. If G = D 8 the argument is identical, using Theo rem 2.2 for k 1 and q 1 .  ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 27 6. Computing p -adic L -functions W e br ieﬂy sk etch the metho d used to compute v alues of p -adic L -functions of 1-dimensional Artin c hara cters. V alues of p -adic L -functions of c hara cters of higher dimension can then b e deduced using (2.11) . The metho d we employ is somewha t similar to , but mor e technical than the construction in [18], and generalizes the one used in [45] to compute v alues of p - adic L -functions of real q ua dratic ﬁelds at s = 1. No te that several modiﬁcations a re necessar y compared to [18] since the construction there is for partial ζ - functions of the ra y class mo dulo p whereas w e need to w ork with partial ζ -functions of a rbitrary cla ss groups. Also , we need to replace p = 2 in a certa in num ber of results b y q = 4. W e lea ve it to the careful reader to check that these c hanges are indeed p ossible, o r refer to the for thcoming article [44] w he r e the details of the actual metho d used will be g iven. Let k b e a to tally rea l n umber ﬁeld and let d = [ k : Q ]. Let χ b e a Q p -v a lued 1-dimensional Artin character of Gal( k /k ). As in Section 2, we let q = p if p is o dd, and q = 4 if p = 2. By clas s ﬁeld theory , χ can b e viewed as a character of a suitable ray class g roup of k . W e denote b y f the conductor of χ , which is an integral idea l o f k , a nd we let g b e the modulus with inﬁnite part g ∞ consisting of a ll inﬁnite places of k , and ﬁnite part g 0 equal to the least common m ultiple of f and q . Let a 1 , . . . , a h be represe ntatives of the classes of the ray cla ss gro up of k mo dulo g , and le t ζ a i ( s ), i = 1 , . . . , h , b e the co rresp onding par tial ζ -functions. Accor ding to [40, Chapter VII, p.526] we hav e, for any isomo rphism σ : Q p → C , h X i =1 σ ◦ χ ( a i ) ζ a i ( s ) = Eul p ( s, σ ◦ χ, k ) L ( s, σ ◦ χ, k ) . Since all ζ a i ( m ) are ra tional for integers m ≤ 0 by lo c. cit. Chapter VII, Co rol- lary 9 .9, we may ident ify them with σ − 1 ( ζ a i ( m )) in Q p . Th us for such m we have (6.1) h X i =1 χ ( a i ) ζ a i ( m ) = Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ) where L ∗ ( m, χ, k ) and Eul ∗ p ( m, χ, k ) are as in (2.7) and (2.8). Now let β 6 = 1 in O k be such that (1) β ≡ 1 (mo d g 0 ) and v ( β ) > 0 for every inﬁnite pla ce v of k ; (2) O k / ( β ) ≃ Z /b Z , where b = N ( β ) is the norm of the principal ideal ( β ). Scaling the measures obtained in [18, Lemme 4.4] for the a i we o bta in Q p -v a lued measures e λ i ( i = 1 , . . . , d ) on Z d p , dep ending o n β and a i , such that for a ll integers m ≤ 0 , ( b m − 1 − 1) ζ a i ( m ) = ( − 1) dm N ( a i ) − m Z ( x 1 · · · x d ) − m d e λ i . The pr o of o f Lemma 4.3 of lo c. cit. shows, contrary to the statement of that lemma, that the measures e λ i are s upp or ted o n ( − 1 + q Z p ) d . Ther efore pulling bac k over m ultiplication by − 1 on Z d p we obtain meas ures λ i on Z d p , supp orted on (1 + q Z p ) d , such that, for the same m , (6.2) ( b m − 1 − 1) ζ a i ( m ) = N ( a i ) − m Z ( x 1 · · · x d ) − m dλ i . 28 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT F o r a ﬁxed x in 1 + q Z p the function s 7→ x s is deﬁned and analy tic on Z p . Since, in g eneral, N ( a i ) will not b e co ngruent to 1 mo dulo q , we do the following. F o r an y integer l a nd an y integral ide a l a of k that is copr ime to p , the fun ction s 7→ ω p ( a ) l hN a i − s is an analytic Z p -v a lued function on Z p , whose v alue equals N ( a ) − m at any integer m with m + l ≡ 0 (mod ϕ ( q )). W e deﬁne the p -a dic par tia l ζ -function o f the cla ss of a i by mapping s in Z p \ { 1 } to ζ p, a i ( s ) =  b s − 1 − 1  − 1 hN ( a i ) i − s Z ′ ( x 1 · · · x d ) − s dλ i , where ′ indicates that we restricted the domain of integration to (1 + q Z p ) d . Let Ψ i be the function deﬁned by the ab ov e integral, that is, for s in Z p , Ψ i ( s ) = Z ′ ( x 1 · · · x d ) − s dλ i . F o r a in Z p with | a − 1 | p ≤ q − 1 , a nd s in Z p , o ne can write a − s = e xp p ( − s log p ( a )), where exp p and lo g p are the p -adic exp onential and logarithm functions resp ectively; and th us the function s 7→ a − s can be expressed as a power series with co eﬃcients in Z p , where the co eﬃcie nt of s m has abso lute v alue at most q − m p m/ ( p − 1) . Developing ( x 1 · · · x d ) − s in this wa y as a p ower ser ies of s a nd using the fact that the measur es λ i hav e bounded norm b y [18, Lemme 4.2 bis], we see that Ψ i ( s ) can also b e expressed as a power series in Q p [[ s ]] where the absolute v alue of the co eﬃcient of s m is a t mos t C i q − m p m/ ( p − 1) for so me C i > 0 . In the same w ay , the function s 7→ hN ( a i ) i − s can b e expressed as a p ow er ser ies in Z p [[ s ]] whos e co eﬃcie nts satisfy the same bounds. Similarly , since the only s o lution for s of b s − 1 = 1 with | s | < q p − 1 / ( p − 1) is s = 1, it follows fro m the p -a dic W eierstrass pre pa ration theorem that the function s 7→ b s − 1 − 1 ca n b e expres sed as the pro duct of a p ow er of p , the po lynomial s − 1, and an inv ertible p ower series in Z p [[ s ]] with the abso lute v alue of the co eﬃcient of s m bo unded by q − m p m/ ( p − 1) . Ther efore ζ p, a i ( s ) can b e ex pressed as the quotient of a p ow er series in Q p [[ s ]] with the absolute v a lue o f the co eﬃcient of s m bo unded by C ′ i q − m p m/ ( p − 1) with C ′ i > 0, and s − 1. If we deﬁne p -adic functions of s , (6.3) L ( l ) p ( s, χ, k ) = h X i =1 ω p ( a i ) l χ ( a i ) ζ p, a i ( s ) ( l mo dulo φ ( q )) , then it follows from (6.1), (6.2), and the equality ζ p, a i ( m ) = ω p ( a i ) m ζ a i ( m ) for a n y int eger m ≤ 0, that (6.4) L ( l ) p ( m, χ, k ) = Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ) for all non-p ositive m such tha t l + m ≡ 0 modulo ϕ ( q ). In particula r, the p - a dic L -function of χ is given b y L p ( s, χ, k ) = L ( − 1) p ( s, χ, k ). Mo reov er, its residue at s = 1 is zero if χ is no n-trivial b eca us e the functions ω p ( a i ) − 1 ζ p, a i ( s ) all hav e the same residue there by [18, Coro llaire on pag e 3 88]. The es timates on the co eﬃcients of the power series expansions ab ov e imply the claims in Theo rem 2.9(1). Throug h the p -a dic W eierstrass prepara tion theorem they also imply that the function in (2 .11) can b e written a s the quotient o f a power ser ies in Q p ( χ )[[ s ]] that c o nv erges for s in C p with | s | < q p − 1 / ( p − 1) , a nd a po lynomial in Q p ( χ )[ s ]. ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 29 W e note in passing that if l is any int eger then L p ( s, χω l p , k ) = h X i =1 ω p ( a i ) l − 1 χ ( a i ) ζ p, a i ( s ) = L ( l − 1) p ( s, χ, k ) , so that we obtain from (6.4) that L p ( m, χω l p , k ) = Eul ∗ p ( m, χ, k ) L ∗ ( m, χ, k ) for all m ≤ 0 with l + m ≡ 1 mo dulo ϕ ( q ), as s tated in Remark 2.13. In or der to compute L ( s, χ, k ) for a ﬁxed s in Z p , it suﬃces to compute ζ p, a i ( s ), for w hich one mainly has to compute the Ψ i ( s ). The measur es λ i can be represented as p ower series in d v ariables , F i ( w 1 , . . . , w d ) = X n 1 ,...,n d ≥ 0 a i,n 1 ,...,n d w n 1 1 · · · w n d d , with (bo unded) co eﬃcients a i,n 1 ,...,n d = Z  x 1 n 1  · · ·  x d n d  dλ i that can b e computed using Shin tani’s metho d [49], which is practica l for ca lc ula- tions if the base ﬁeld k is Q or real qua dratic. On the other hand, the function h on Z p deﬁned by x 7→ ( x − s if x is in 1 + q Z p , 0 otherwise , is contin uous and th us admits a Mahler ex pansion P n ≥ 0 c s,n  x n  , where the co eﬃ- cients satisfy c s,n → 0 when n → ∞ and ca n b e easily computed r ecursively . Then we have Ψ i ( s ) = Z h ( x 1 ) · · · h ( x d ) dλ i = X n 1 ,...,n d ≥ 0 a i,n 1 ,...,n d c s,n 1 · · · c s,n d . Thu s, by computing enough terms in the above sum, we can get a go o d approx- imation of the integral, and therefor e, putting everything together, of the p -adic L -function. In order to apply this w e mak e (2.10) explicit in the ca ses we ar e in terested in, writing L - functions of representations for the L -functions of the c orresp ond- ing characters. In particula r, w e write the classical L -functions of the ir reducible 2-dimensional representations of S 3 and D 8 in Examples 5.3 and 5.5, and the irre - ducible 2 -dimensional representations W a with a 6 = 1 o f S 3 × Z / 3 Z in Example 5 .7, as the L -functions o f Ab elian characters ov er quadratic ﬁelds. F or la ter use we also note how to write s ome o f them in terms of ζ -functions of certain ﬁelds. W e take the ﬁeld o f co eﬃcients E to b e C in the remainder of this section, and denote the trivial 1-dimensional representation of any g roup G by 1 G . Example 6.5 . Let Alt S 3 be the 1-dimensional representation of S 3 through the sign character, and V th e ir reducible 2-dimensional repr esentation. Since Ind S 3 A 3 ◦ Ind A 3 { e } (1 { e } ) ∼ = Ind S 3 { e } (1 { e } ) ∼ = 1 S 3 ⊕ Alt S 3 ⊕ V 2 and Ind S 3 A 3 (1 A 3 ) ∼ = 1 S 3 ⊕ Alt S 3 it follows that Ind S 3 A 3 ( V ′ ) ∼ = V for either non-trivial 1 - dimensional repre s ent ation V ′ of A 3 . Applying this to the situation of Exa mple 5.3 we see that L ( s, V , Q ) = L ( s, V ′ , q ). Similarly , if H is any subgroup of S 3 of 30 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT order 2, then Ind S 3 H (1 H ) ∼ = 1 S 3 ⊕ V . In the situation o f Example 5.3 this means that ζ c ( s ) = ζ Q ( s ) L ( s, V , Q ). Example 6.6. If D 8 is as in Example 5.5, V a,b for a, b = ± 1 the 1-dimensional representation o f D 8 where σ a cts as m ultiplication b y a and τ by b , and V the irreducible 2-dimensio na l repres ent ation, then for any subgro up H Ind D 8 H ◦ Ind H { e } (1 { e } ) ∼ = ( ⊕ a,b = ± 1 V a,b ) ⊕ V 2 . If we take H = h τ , σ 2 i then Ind H { e } (1 { e } ) ∼ = ⊕ a,b = ± 1 W a,b with W a,b the 1-dimens ional representation o f H where σ 2 acts as multiplication b y a and τ by b . Because Ind D 8 H ( W 1 ,b ) ∼ = ⊕ a = ± 1 V a,b we hav e Ind D 8 H ( W − 1 , 1 ) ∼ = Ind D 8 H ( W − 1 , − 1 ) ∼ = V , so with notation as in Fig ure 5.1 we get L ( s, V , Q ) = L ( s, W − 1 , 1 , q 1 ) = L ( s, W − 1 , − 1 , q 1 ). Similarly , Ind D 8 h τ i (1 h τ i ) ∼ = ( ⊕ a = ± 1 V a, 1 ) ⊕ V so that ζ k 1 ( s ) = ζ q 1 ( s ) L ( s, V , Q ) as well. Remark 6.7. If H = h σ i in Example 6.6 then Ind H { e } (1 { e } ) ∼ = ⊕ 4 m =1 U i m with U a the 1-dimensional representation of h σ i where σ acts as multiplication by a . Now Ind D 8 H ( U a ) ∼ = V a, 1 ⊕ V a, − 1 when a = ± 1 so that Ind D 8 H ( U a ) ∼ = V when a 6 = ± 1, a nd in Ex ample 5 .5 we also hav e L ( s, V , Q ) = L ( s, U i , q 2 ) = L ( s, U − i , q 2 ). Example 6. 8. If G = S 3 × Z / 3 Z a nd H = A 3 × Z / 3 Z then Ind G H ◦ Ind H { ( e, 0) } (1 { ( e, 0) } ) ∼ = W ⊕ W 2 1 ⊕ W 2 ζ 3 ⊕ W 2 ζ 2 3 with W the dir ect sum of the 1-dimens ional represe ntations of G and the W a as in Example 5.7. W rite Ind H { ( e, 0) } (1 { ( e, 0) } ) ∼ = ⊕ a,b V a,b where, for a a nd b cubic ro ots of unity , V a,b is the 1-dimensiona l representation of H on whic h ( σ m , n ) acts as m ultiplication by a m b n . Since Ind G H ( ⊕ b V 1 ,b ) ∼ = W we ﬁnd b y considering the actio n of ( e, 1) th at Ind G H ( V a,b ) ∼ = W b when a 6 = 1. In particular, in the notation o f Example 5 .7, we have L ( s, W b , Q ) = L ( s, V a,b , q ) since q = k H . It is easy to see that the co eﬃcients of p − s in the E uler factors of L ( s, W ζ 3 , Q ) and L ( s, W ζ 2 3 , Q ) ar e conjugate, but since not all of them can b e real it is not po ssible to e xpress those functions in ter ms o f ζ -functions of subﬁelds o f k . 7. Examples In this s ection we describ e the evidence for Conjecture 3 .18 in the cas e of prim- itive idemp otents π in E [Gal( k / Q )] as in Ex amples 5.3, 5 .5 and 5.7, with E = Q in the ﬁrst t wo and E = Q ( ζ 3 ) in th e last. In the ﬁrst tw o cases par t (1) of the conjecture is kno wn by Prop ositio n 5.9, but in the la st ca se our ev ide nc e for this part is numerical; the same holds for parts (2) thr ough (4) in all case s . W e would like to stress that the n umerical veriﬁcation of pa rt (4) of the conjecture actually pr oves this part for all ca ses that we considere d since we can chec k that a element of E ⊗ Q p is a unit o f that ring by computing a close enough a pproximation of it. F o r the numerical ca lculations o f the p -adic regulator , R n,p ( M E π ), w e assumed that the syntomic reg ulator map on the subg roup of K 2 n − 1 ( k ) Q describ ed in Zagier’s Conjecture 4.5 is given by the e P σ n,p (as in Remar k 4.15). Out of the many p ossible choices fo r P n,p ( z ) as descr ibe d in (4.1 2) we use d P n,p ( z ) = Li n,p ( z ) − log n − 1 p ( z )Li 1 ( z ) / n ! = Li n,p ( z ) + log n − 1 p ( z ) log p (1 − z ) /n ! ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 31 since it is relatively simple from a c o mputational p oint of view. The calculations of the P n,p ( z ) for the br anch with lo g p ( p ) = 0 were done in versions 2.11-7 , 2.12 -19 and 2 .12-21 o f [52], using an implemen tation o f the a lg orithm describ ed in [6]. Note that the primitive idemp otents in Examples 5.3 and 5.5 are unique up to conjugation in Q [ G ] since they all corresp ond to π 2 . But in Example 5.7 replacing ζ 3 with ζ 2 3 changes a primitive idemp o tent cor resp onding to π 3 int o one corr esp onding to π 4 , and conv ersely . Bec ause th e sa me holds for the classica l and p -a dic L - functions (with v alues in E ⊗ Q C and E ⊗ Q Q p resp ectively), as well a s the Euler factor for p , the conjectures for primitive idemp otents cor resp onding to π 3 and π 4 are a ctually equiv a lent . In Examples 7.1, 7.4 and 7.5 we hav e E = Q so tha t we identif y E ⊗ Q C with C , etc., a nd use nota tion a s in Remark 3.17. Example 7. 1. Let k b e a totally r eal S 3 -extension of Q . Use no tation as in Ca se 1 of Example 5.3, and let π b e the pr imitive ide mp otent cor resp onding to π 2 of that example. Since E = Q we shall write M π instead of M E π in all no tation referring to Co njecture 3 .18. Since π K 2 n − 1 ( k ) Q = π 2 K 2 n − 1 ( c ) Q by Remark 3.2 2 we computed the K -theory of c using the (numerical) metho ds of Section 5 and tr ie d to reco gnize the num b er (7.2) e ( n, M π ) = L ( n, χ π , Q ) · D ( M π ) 1 / 2 , ∞ /R n, ∞ ( M π ) for n = 3 and 5 as an elemen t of Q ∗ by employing the same metho ds a s describ ed in Section 5 for n umber ﬁelds , where L ( n, χ π , Q ) was computed as ζ c ( n ) /ζ Q ( n ) (see E xample 6.5) using Pari-GP [53]. W e succee de d in all cases, and then veriﬁed nu merica lly for p = 2, 3, 5, 7 a nd 11 if L p ( n, χ π ω 1 − n p , Q ) 6 = 0 a nd (7.3) e ( n, M π ) · Eul p ( n, χ π , Q ) · R n,p ( M π ) D ( M π ) 1 / 2 ,p · L p ( n, χ π ω 1 − n p , Q ) − 1 in Q ∗ p was equal to 1. W e put our re sults for the four tota lly r eal S 3 -extensions k o f Q with smallest dis - criminants in T ables 1 thro ugh 4 . No te that by Lemma 3.7 R n,p ( M π ) /D ( M π ) 1 / 2 ,p is in Q p and the same holds for L p ( n, χ π ω 1 − n p , Q ) by Lemma 3.15, so that (ap- proximations of ) those num ber s can be easily repres ented in our tables . W e ha ve denoted an element ( a 0 + a 1 p + a 2 p 2 + · · · ) × p s in Q ∗ p with all a j in { 0 , . . . , p − 1 } and a 0 6 = 0 by ( a 0 .a 1 a 2 · · · ) p × p s , writing A to repres e nt 10 when p = 1 1 . F o r the p -adic regulator s we computed each P n,p ( z ) up to O ( p L ( p ) ) with p L ( p ) approximately equal to 1 0 30 . The rele v ant par t of R n,p ( M π ) /D ( M π ) 1 / 2 ,p has b een given in the tables whenev er it ﬁtted. F or the v alues of the p - adic L -function w e can only pr ov e that the relative error is at most 1 + O ( p M ( p ) ), where M (2) = 72, M (3) = 47 , M (5) = 3 2, M (7) = 26 and M (11) = 22. This pro ves that L p ( n, χ π ω 1 − n p , Q ) does no t v anish; together with the veriﬁcation that (7 .2) equals 1 numerically this prov es part (4) of Co njecture 3.1 8. The v alue that we found for (7.3) was 1 + O ( p N ( p ) ) with N (2) ≥ 82, N (3) ≥ 52 , N (5) ≥ 36 , N (7) ≥ 27 and N (11) ≥ 24 , g iv ing numerical evidence for parts (2) and (3) of the conjecture, but also sugg e sting that the relative precision of the v alue of the p -adic L -function is slightly higher than w e ca n prove, justifying the higher precision given in the tables. Similar calculations w ere carried out for the next four suc h S 3 extensions and the same primes, with very similar res ults. 32 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT The other examples pr o ceed mostly a lo ng the s ame lines as Example 7.1. The v alues of L ( n, χ π , Q ) in E x amples 7.4 and 7.5 b elow were computed using the for- m ula at the e nd of E xample 6.6, but in Exa mple 7.6 the v alues of L ( n, χ E π , Q ), with E = Q ( ζ 3 ) for ζ 3 a primitive cub e ro ot of unit y , w ere computed using the algorithms describ ed in [24] and its asso ciated pr ogra m. In Examples 7.4 and 7.5 we determined if e ( n, χ π ) (as in (7.2)) w as in Q ∗ using the metho ds of Section 5, and for Example 7.6 w e wrote e ( n, M E π ) − 1 = 1 ⊗ a + ζ 3 ⊗ b in E ⊗ Q R a nd then recognized a and b as integers divided by pr o ducts of some r elatively small primes. In ea ch ca se we then veriﬁed for p = 2 , . . . , 11 if (7.3) (or its equiv alent) was equal to 1 in Q p or, for Exa mple 7 .6, in Q p ( ζ 3 ) ⊗ Q Q = Q p ( ζ 3 ). The precis io n fo r P n,p ( z ) and the p -adic L -functions in Examples 7.4 and 7.5 was a s in E x ample 7.1. In Example 7.6 all P n,p ( z ) were c o mputed up to O ( p L ( p ) ) with p L ( p ) approximately 10 16 since in this case the chec k if the eq uiv alent of (7.3) is 1 is done in Q ( ζ 3 ) ⊗ Q Q p so tw o c o eﬃcients are check ed; the v alues of the p -a dic L -function in this ca se, also in Q ( ζ 3 ) ⊗ Q Q p , w ere determined up to multiplication by 1 ⊗ (1 + O ( p M ( p ) )) with M ( p ) as in Example 7.1. Example 7. 4. Let k b e a totally real D 8 -extension o f Q as in Case 1 of E xample 5.5, and use nota tion as in that example. W e a gain write M π instead o f M E π since E = Q and co nsider odd n ≥ 2. Because πK 2 n − 1 ( k ) Q = π 2 K 2 n − 1 ( k 1 ) Q we computed K 2 n − 1 ( k 1 ) Q nu merica lly using the metho ds of Section 4. Our results with n = 3 and 5 for the ﬁr st fo ur such extensions when ordere d accor ding to the discr iminant are in T ables 5 through 8. The v alue that w e found for (7.3) in this ca se was 1 + O ( p N ( p ) ) wher e N (2) ≥ 7 6, N (3) ≥ 5 2 , N (5) ≥ 3 4, N (7) ≥ 29 a nd N (11) ≥ 22 . Such calculations w ere a lso c arried out for the nex t four suc h extensions, with similar r esults. Example 7.5. Let k b e a CM Galois extension of Q with Gal( k / Q ) ∼ = D 8 as in Case 2 of Ex ample 5.5, and use notation as in that example. W e again write M π instead of M E π since E = Q , and c o nsider even n ≥ 2. Again π K 2 n − 1 ( k ) Q = π 2 K 2 n − 1 ( k 1 ) Q , so we computed K 2 n − 1 ( k 1 ) Q nu merica lly using the metho ds of Sec- tion 4. W e put our results at n = 2 and 4 for the ﬁrs t four such extensions when ordered by the v alue o f the (p os itive) discr iminant in T ables 9 thr ough 12. The v alue that we found for (7.3) in this case was 1 + O ( p N ( p ) ) where N (2) ≥ 79, N (3) ≥ 5 3 , N (5) ≥ 35, N (7) ≥ 27 and N (11 ) ≥ 25. Such calculations w ere also carried out fo r the next four such extensions , with simila r results. Example 7.6. Let k be a totally real S 3 × Z / 3 Z extension of Q , as in Ca s e 1 of Example 5.7. W e use notation as in that ex a mple, taking E = Q ( ζ 3 ) for a primitive cubic ro o t of unit y ζ 3 . A diﬀerent iden tiﬁcation of Gal( k / Q ) with S 3 × Z / 3 Z might int erchange π 3 and π 4 as well as corr esp onding primitive idempotents, but, a s we noted b efor e, the v alidit y of the conjecture for either is equiv alent. Since π K 2 n − 1 ( k ) E = π 3 K 2 n − 1 ( s 1 ) E we co mputed K 2 n − 1 ( s 1 ) Q as b efore. Also , in order to s pee d up the calculations of the r e gulator w e note that the a ction of ( e, 1 ) and ( σ 2 , 0 ) on s 1 = k h ( σ, 1) i is the same, so tha t π = π 3 π H 1 and (1 − π 1 )(( e, 0) + ζ 3 ( σ , 0) + ζ 2 3 ( σ 2 , 0 )) / 3 = (( e, 0) + ζ 3 ( σ , 0) + ζ 2 3 ( σ 2 , 0 )) / 3 give the sa me result when applied to K 2 n − 1 ( s 1 ) E . W e put our re s ults for the ﬁrst four s 1 , when ordered accor ding to their discrimina nt, for n = 3 and 5 in T ables 13 ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 33 through 1 6 . In T a ble 16 the n umber e (5 , M E π ) − 1 is g iven b y − 2 5 · 3 10 · 5 − 2 · 13 4 · (7 + 3 ζ 3 ) · (68 − 43 ζ 3 ) · (15 47202 − 160 3 487 ζ 3 ) · ( a + bζ 3 ) with a = 8 4822 55892 311139091186217686741233 and b = 352744 0500 0 58817421018094757947617 , presumably b ecause we exp erimentally found a subgro up of K 2 n − 1 ( s 1 ) Q that is not close to actually being K 2 n − 1 ( s 1 ). In all cases L p ( n, χ π ⊗ ω 1 − n p , Q ) is a unit in E ⊗ Q Q p , and when we write the equiv alent of (7.3) in the fo rm 1 ⊗ α + ζ 3 ⊗ β then we ﬁnd that α − 1 and β a re O ( p N ( p ) ) with N (2) ≥ 50, N (3) ≥ 24, N (5 ) ≥ 20, N (7) ≥ 14 and N (1 1 ) ≥ 1 2, with in this case the pr ecision b ounded by that of the R n,p ( M E π ). Since e ( n, M E π ), D ( M E π ) 1 / 2 ,p and E ul p ( n, χ π ⊗ id , Q ) a re units in E ⊗ Q Q p this also shows that R n,p ( M E π ) is a unit in E ⊗ Q F , as conjectured in Conjecture 3.18(4). Such calculations were also carr ied out for the nex t four such s 1 with similar results. Remark 7.7. By Pr op osition 4.17 and Remark 3.21(1), the conjecture for π in Example 7.1 is equiv alent to Conjecture 2.17 for the ﬁe ld k 1 in the ex ample. The same statement holds for π and k 1 in Example 7.4 b elow. Remark 7.8. In the tables we hav e also included the v alue of the consta n t C ( χ π ⊗ id) = C ( χ ∨ π ⊗ id) in the functional equation (3.2 5) of L ( s, χ π ⊗ id , Q ) to make po ssible a compar ison betw een its prime facto rs and those of e ( n, M π ). Namely , fo r the D ( M E π ) 1 / 2 , ∞ used in the calculations, the e n in (3.26) (which in our case a r e independent of n ) are as follows. T able e n T able e n T able e n T able e n 1 3 5 4 9 4 13 1 ⊗ 9 + ζ 3 ⊗ 9 2 3 6 4 10 4 14 1 ⊗ 9 + ζ 3 ⊗ 9 3 3 7 4 11 − 4 15 1 ⊗ 0 + ζ 3 ⊗ 9 4 3 8 4 12 4 16 1 ⊗ 0 + ζ 3 ⊗ ( − 9) As ca n b e seen fr om these v alues and T ables 1-1 6, in the num b er δ n e n (( n − 1 )! / 2) m C ( χ ∨ π ⊗ id) n − 1 e ( n, M E π ) of (3.27) the la rger prime factors o ccurring in e ( n, M E π ) ar e often cancelled by thos e of C ( χ ∨ π ⊗ id ) n − 1 . This sugg e sts that the for mu lation of Beilinson’s conjecture a t s = 1 − n normally inv olves simpler prime factor s in E ∗ than the formulation at s = n in Conjecture 3.18(1), with the remaining complica ted factors in our examples quite p os s ibly due to an awkw ard choice o f basis fo r K 2 n − 1 ( M E π ). 34 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT References [1] D. Barsky . F onctions zˆ eta p -adiques d’une classe de ray on des corps de nom bres totalement r´ eels. 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In Arithmetic algeb r aic ge ometry (T exel, 1989) , pages 391–430. Birkh¨ auser Boston, Boston, MA, 1991. Dep ar tmen t of Ma thema tics, Ben -G urion Un iversity of the Negev, P.O.B. 653 , Be’er- Shev a 84105, Israel Dep ar tmen t of Pure Ma thema tics, University of Sheffield, Hicks Building, Hou n sfield R oad, Shef field S3 7RH, United Kingdom F acul teit Exacte Wetenschappen, Afdeling Wiskunde, Vrije Universiteit, De Boele- laan 10 81a, 10 81 HV Amsterdam, The Netherlan ds Institut Camille Jordan, Universit ´ e de L y on , Universit ´ e L yon 1, CNRS – UM R 5 2 08, 43 bl vd du 11 Novembre 191 8, 6962 2 Villeurba n ne Cedex, France ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 37 T able 1. Splitting ﬁeld of x 3 − x 2 − 3 x + 1, where C ( χ π ) = 2 2 · 37. n = 3 e (3 , M π ) − 1 = 2 − 2 · 3 − 2 · 37 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0001010 100100000001 0000101111100100010101110100110010110110011000000) 2 × 2 0 3 (1 . 0100121 212022110212 0000110212112110020120110012120021102 1212) 3 × 3 4 5 (4 . 2320143 330214023104 113344110103131003140) 5 × 5 6 7 (6 . 3543043 014124124154 50326016336635) 7 × 7 6 11 (2 . 6216123 5 A 928 A 3423563 A 788 8 A ) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0000001 010011100010 0001100100011111010110000101001101110011000101000) 2 × 2 2 3 (1 . 1202122 002221002110 112100012002001211111011101010220 ) 3 × 3 0 5 (4 . 1104002 444023244223 3024131040014140) 5 × 5 0 7 (5 . 2351636322650 1261362543533110) 7 × 7 0 11 ( A. 9542728 A 692 401225487 A 278 ) 11 × 11 0 n = 5 e (5 , M π ) − 1 = 2 4 · 3 2 · 5 − 2 · 37 4 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0000101 011011001110 1000011101001000001100000100010000011011110011101) 2 × 2 6 3 (1 . 2001110 220201020112 02110020222210102001020101122110122022) 3 × 3 12 5 (3 . 2142103 424002243031 22204142313310423002) 5 × 5 8 7 (1 . 21030044 053554501231 24426604231) 7 × 7 10 11 (1 . 2069 AA 60760 924283465546 5) 11 × 11 10 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0000010 111100111000 1111000010111110100101010001010001001100111100011) 2 × 2 2 3 (1 . 2022122 002112102001 110211120022002211212022201001210 ) 3 × 3 0 5 (3 . 3440442 211312114142 0412040113130342) 5 × 5 0 7 (4 . 6600156551362 2316645262615350) 7 × 7 0 11 (1 . 1 A 26918431 A 0920009 9 A 81 A 7) 11 × 11 0 T able 2. Splitting ﬁeld of x 3 − 6 x − 2, where C ( χ π ) = 2 2 · 3 3 · 7. n = 3 e (3 , M π ) − 1 = − 2 − 2 · 3 4 · 7 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1000000 111000011010 1010101111101011001000101111010100001000101111010) 2 × 2 0 3 ( 2 . 112202001122 0001210112202100011222122000102121211020022012) 3 × 3 4 5 (3 . 4232302 100220101433 222440213432224202143) 5 × 5 6 7 (4 . 2544363 600510010013 523314315122) 7 × 7 7 11 (5 . 4815695 41633 A 6875525 A 00 911) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0111110 111001100110 0110111100111100010100000100110111100101110110010) 2 × 2 2 3 (1 . 1211000 101210110112 021221022022211101101102021011220 ) 3 × 3 0 5 (2 . 1300002 012230201122 1412322140400441) 5 × 5 0 7 (4 . 5551015241624 5405156545556000) 7 × 7 2 11 (1 . 14 A 215770 33 A 19227344806 A ) 11 × 11 0 n = 5 e (5 , M π ) − 1 = − 2 4 · 3 14 · 5 − 2 · 7 4 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1010110 110000010110 0101100100110101101111111000110011100110000011111) 2 × 2 6 3 (2 . 1201222 220112100121 10212011212210211022212221020022201) 3 × 3 14 5 (4 . 0402101 420443343344 12003042243311012202) 5 × 5 8 7 (3 . 2403415 331415026502 116243110353) 7 × 7 9 11 (7 . 1609640 965036435375 18492) 11 × 11 10 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0101011 100001011101 1111101010101111011001011010111110100101100010111) 2 × 2 2 3 (1 . 2002000 121100001212 101001020022211022201122100111000 ) 3 × 3 0 5 (4 . 4431110 233344212121 2402410333144343) 5 × 5 0 7 (3 . 4152603036346 5662002262442000) 7 × 7 0 11 (2 . 6873493 21 A 27130485753 530) 11 × 11 0 38 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT T able 3. Spli tting ﬁeld of x 3 − 4 x − 1, where C ( χ π ) = 229. n = 3 e (3 , M π ) − 1 = − 2 − 9 · 3 − 2 · 229 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0111010 010101000000 1111100101001000110110101000001001011100111100010) 2 × 2 0 3 (2 . 0120122 110101210110 2110122002101221210011000222020021102 1221) 3 × 3 4 5 (4 . 3433241 433301132322 003123241441021322342) 5 × 5 6 7 (6 . 2624016 345625053536 33061004510116) 7 × 7 6 11 (3 . 6717 A 2040 148108526534 2054) 11 × 11 7 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0110000 101100110111 0100010111011001000000110000010000100110111111101) 2 × 2 3 3 (2 . 2120211 202111122100 220011211121110002020101200021220 ) 3 × 3 0 5 (3 . 0341433 224143344140 4241012442104412) 5 × 5 0 7 (3 . 2403304413252 3034433054506150) 7 × 7 0 11 (9 . 21 A 61 A 480 A 10 801760893710 ) 11 × 11 1 n = 5 e (5 , M π ) − 1 = − 2 − 8 · 3 6 · 11 · 229 4 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1111011 010101001101 1010001011100111001101011001101110111000000010001) 2 × 2 5 3 (2 . 2221120 210120100012 01220011021022211112022112001020100) 3 × 3 16 5 (4 . 3414223 144333221124 000410221113420102) 5 × 5 10 7 (2 . 56120366 056462666352 64251053110) 7 × 7 10 11 (8 . 5 A 9220278 400758 A 0193 AA 6) 11 × 11 11 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0101000 011101010001 0011010011100110010001101000101101100101100100011) 2 × 2 3 3 (2 . 1121000 022222000210 222202101202022011021200021010220 ) 3 × 3 0 5 (4 . 4201113 234211033234 1130324344440134) 5 × 5 0 7 (1 . 2503224516465 6402236525435500) 7 × 7 0 11 ( 5 . 375705346 AA 0810 959728302) 11 × 11 0 T able 4. Splitting ﬁeld of x 3 − 6 x 2 + 2, where C ( χ π ) = 2 2 · 3 4 · 5. n = 3 e (3 , M π ) − 1 = 2 − 2 · 3 6 · 5 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1111001 011000000101 1011110010111010011100101111011011000111111110001) 2 × 2 0 3 (2 . 1200201 000000121020 10221000201210120200202222221111002100) 3 × 3 6 5 (3 . 3432222 440014423344 440441124111231400414) 5 × 5 4 7 (5 . 6116512 666226565550 64166304513241) 7 × 7 6 11 (2 . 805 A 83442 65760 A 69 A 249082 8 A ) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 1010000 010011011111 0111110101101001001110101011111100111101100001111) 2 × 2 2 3 (1 . 1120211 121012011012 202202010100211210110122222201210 ) 3 × 3 0 5 (2 . 1324231 323301120113 3333143412022134) 5 × 5 0 7 (3 . 0012344214261 3541412624063510) 7 × 7 0 11 ( 2 . 2725 A 17844929 880 A 8412281) 11 × 11 0 n = 5 e (5 , M π ) − 1 = − 2 5 · 3 18 · 5 2 · 324762301 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0001010 001101100111 0000011110010000100001101100111101000101001001110) 2 × 2 7 3 (1 . 0200220 011011111211 112101212120212201200221122102001) 3 × 3 18 5 (1 . 3211002 211131421044 12032003401031323110) 5 × 5 7 7 (3 . 55530636 653054056501 05042513546) 7 × 7 10 11 ( A. 568 607725325302 A 5663 A 746) 11 × 11 10 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 1011110 101001100111 1001000001111111000110110101100010100000000001010) 2 × 2 2 3 (1 . 0002101 112212121121 001002201100122111211210111102110 ) 3 × 3 0 5 (2 . 3033413 021312032241 0240113023441000) 5 × 5 0 7 (4 . 6204210112233 2630656130062420) 7 × 7 0 11 ( 5 . 491153185997 A 48326862 A 46) 11 × 11 0 ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 39 T able 5. Spli tting ﬁeld of x 4 − x 3 − 3 x 2 + x + 1, where C ( χ π ) = 5 · 29. n = 3 e (3 , M π ) − 1 = 2 − 6 · 3 − 1 · 29 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1111101 011110001110 0011000110000111000001111111101101000111001001001) 2 × 2 2 3 (1 . 2012202 111200101100 1020012022101210110112101120121220221 0212) 3 × 3 5 5 (2 . 21221421224100 432231134432324224211113143) 5 × 5 3 7 (2 . 2605241 466266246050 01621634435032) 7 × 7 6 11 (2 . 7469785 544497786253 1 A 25026) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0101001 010011100010 1001000111001101011100000001100110101111010001100) 2 × 2 2 3 (1 . 1011210 202202201201 020222100121100002100020220220021 ) 3 × 3 0 5 (4 . 3021032 442342201414 2420103340302330) 5 × 5 0 7 (1 . 1635506314325 1352506540201034) 7 × 7 0 11 (9 . 9 A 7720 A 981020 A 9 A 32071 790) 11 × 11 0 n = 5 e (5 , M π ) − 1 = 2 − 4 · 3 2 · 5 2 · 29 4 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0110110 100110110001 1000000000011101110110010001010001100111111101001) 2 × 2 8 3 (1 . 0002011 102111201211 1200120120112211122102121111120022021 22) 3 × 3 12 5 (1 . 0302040 011301341231 30241412102302410321) 5 × 5 7 7 (5 . 62133315 453444316136 00254300115) 7 × 7 10 11 (9 . 9841445960878 9 A 37989 AA 86) 11 × 11 10 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0111110 010110010111 0101001110010110011011010001011101111000000111001) 2 × 2 2 3 (1 . 1111012 010001010110 000200000021011010010220212100211 ) 3 × 3 0 5 (4 . 0134142 141114000202 4312030010110013) 5 × 5 0 7 (4 . 1003213463255 3222245313526015) 7 × 7 0 11 (8 . 6637671 480756 A 9916284 377) 11 × 11 0 T able 6. Spli tting ﬁeld of x 4 − 2 x 3 − 3 x 2 + 4 x + 1, where C ( χ π ) = 2 2 · 3 2 · 11. n = 3 e (3 , M π ) − 1 = 2 − 2 · 3 2 · 11 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1100111 111110011001 1011000011011110100000010001100010000111000001110) 2 × 2 5 3 (1 . 1012221 122001111200 0212220021002200202112120010111202221 2121) 3 × 3 2 5 (1 . 3213042 203212141302 102134014142431100344) 5 × 5 6 7 (1 . 4444342 055331115320 05253505615656) 7 × 7 6 11 (9 . 61613555149153 993505670160) 11 × 11 5 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 1111010 110100100110 0111001001100001010010111011100011010101000010001) 2 × 2 4 3 (1 . 1221211 120021010011 021120022021022220202012220120020 ) 3 × 3 0 5 (1 . 0312343 242124131121 0024333243134200) 5 × 5 0 7 (1 . 2051510450153 5164355503412253) 7 × 7 0 11 (4 .A 19905 A 9766 7183 A 61256312 ) 11 × 11 0 n = 5 e (5 , M π ) − 1 = 2 7 · 3 10 · 5 − 2 · 11 7 · 151 · 1389251 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1010100 010001100100 1110000000110011010000011010001100100101100100011) 2 × 2 16 3 (2 . 0002001 102001121012 1200021121000100102202022102012021120) 3 × 3 10 5 (3 . 0431243 401011420413 02443131432222411101) 5 × 5 8 7 (3 . 10160421 211515534425 03432550426) 7 × 7 10 11 (4 .A 244 AA 713986 547114383) 11 × 11 13 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0101100 111100010010 0100110110111001110010101101110101100000001001101) 2 × 2 4 3 (1 . 2200111 010220200001 122120000000211211201112212022102 ) 3 × 3 0 5 (4 . 3111122 441240321001 0123414333414223) 5 × 5 0 7 (2 . 6346365164142 4006321031014001) 7 × 7 0 11 ( 2 . 054143779893 294 A 023 A 2760) 11 × 11 1 40 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT T able 7. Splitting ﬁeld of x 4 − 6 x 2 + 6, where C ( χ π ) = 2 7 · 3 2 . n = 3 e (3 , M π ) − 1 = − 2 8 · 3 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1111111 011010111011 0001001001000100111010011011111000101011110110110) 2 × 2 10 3 (2 . 0221120 120201210000 1201022202121010112000021122011222211 0100) 3 × 3 2 5 (2 . 0412342 403311100123 13440303434111221114) 5 × 5 7 7 (4 . 1261264 126014430665 06634321456330) 7 × 7 6 11 (1 . 508753 A 40 028580 A 83265 A 63 97) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0011100 111101010011 0000110000111111000001101110001010000100110100110) 2 × 2 2 3 (1 . 1102020 120122110111 012211100112220010211120222001022 ) 3 × 3 0 5 (3 . 2410242 342103032233 0312322041041320) 5 × 5 1 7 (3 . 5342155311155 2403452502135351) 7 × 7 0 11 (9 . 993168 A 73 395501045 A 926 A 6 ) 11 × 11 0 n = 5 e (5 , M π ) − 1 = − 2 25 · 3 10 · 5 − 1 · 11 · 37 · 180097 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0100001 111010100011 1110110010101100000101100100001110011010100110111) 2 × 2 27 3 (1 . 2011021 002212120212 0201021000120210110011111202222221202) 3 × 3 10 5 (4 . 1332101 022221332114 3100244321412404344) 5 × 5 9 7 (1 . 65422600 451043113150 15635541014) 7 × 7 10 11 (4 . 3578696 869289212797 6707) 11 × 11 11 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 0010010 010111111100 1101001011101011111110110100110100000111111010001) 2 × 2 2 3 (1 . 0101222 010121001211 100001210022220120122121200102122 ) 3 × 3 0 5 (2 . 4433424 021314403334 0330341233100231) 5 × 5 0 7 (2 . 4361323134355 5620664002413300) 7 × 7 0 11 ( A. 957125591 551731675713 855) 11 × 11 0 T able 8. Splitting ﬁeld of x 4 − 6 x 2 − 4 x + 2, where C ( χ π ) = 2 5 · 17. n = 3 e (3 , M π ) − 1 = − 2 2 · 3 − 2 · 17 2 p R 3 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1011100 011011101110 0110001011111110001011110100001100011100001101001) 2 × 2 7 3 (1 . 1001122 110011102222 210022111010102000101002122001210120111212) 3 × 3 4 5 (2 . 1330133 431400143111 041343243044132341000) 5 × 5 6 7 (1 . 2513660 216406562155 54304534633666) 7 × 7 6 11 (6 . 8781153 571182 A 7 A 3 A 525785 90) 11 × 11 6 p L p (3 , χ π ω − 2 p , Q ) 2 (1 . 0110010 100001101111 0000111011100100110110110111011000100000101100110) 2 × 2 5 3 (2 . 2111111 121010122120 102220010120102202210101102021112 ) 3 × 3 0 5 (2 . 0301013 124332134224 4201240202313243) 5 × 5 0 7 (2 . 4565354652534 5623002012601006) 7 × 7 0 11 (1 . 5054539 309942 A 8686113 521) 11 × 11 0 n = 5 e (5 , M π ) − 1 = − 2 14 · 3 2 · 5 − 2 · 17 4 p R 5 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0110100 110011110001 1100000011101100011011111111011011010000000100111) 2 × 2 19 3 (1 . 0022021 200111200200 1100221001101012201000212010200200210 01) 3 × 3 12 5 (4 . 4112222 132244114213 1444023001440141102) 5 × 5 9 7 (4 . 64256566 210002326345 42266362653) 7 × 7 10 11 (5 . 3066952 6 A 542413 A 1555910 6) 11 × 11 10 p L p (5 , χ π ω − 4 p , Q ) 2 (1 . 1000011 100110010000 1010000010001000011100111101011000000110111100101) 2 × 2 5 3 (2 . 1100001 122200222122 120000002011201000121010120122102 ) 3 × 3 0 5 (1 . 0031343 413141143440 3030110103232430) 5 × 5 1 7 (4 . 5140454150405 2312152404613255) 7 × 7 0 11 ( 2 .A 433150613007 27 A 39739622) 11 × 11 0 ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 41 T able 9. Splitting ﬁeld of x 4 − 2 x 3 + 5 x 2 − 4 x + 2, where C ( χ π ) = 2 3 · 17. n = 2 e (2 , M π ) − 1 = − 17 p R 2 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1111000 001111100010 0011010100111010001000101111111000010111001000001) 2 × 2 8 3 (1 . 2001010 102220112101 111101220000000012020211022222012112010122) 3 × 3 4 5 (2 . 01203020434342 403202140220421203111323421) 5 × 5 4 7 (4 . 1140305 454665411023 106103225364300) 7 × 7 4 11 (9 . 0746676 49 A 54466861880 828 A 41) 11 × 11 4 p L p (2 , χ π ω − 1 p , Q ) 2 (1 . 1001110 001000100111 0011010011101110110111100101101100011101101010011) 2 × 2 6 3 (1 . 2211021 011200210000 101010021221011201210202220121211 ) 3 × 3 0 5 (4 . 3300340 310111240210 3034043124123020) 5 × 5 0 7 (6 . 5565525440030 2602216266140200) 7 × 7 0 11 (4 . 84968 AAA 0466 783629843316 ) 11 × 11 0 n = 4 e (4 , M π ) − 1 = 2 4 · 3 2 · 17 3 p R 4 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0111101 010101111010 0100110010011000001110001101011101000010011010010) 2 × 2 15 3 (2 . 0202020 222222221201 1121000222200020002221002201220201002211) 3 × 3 10 5 (3 . 1211414 344410311000 40041101320123332342) 5 × 5 8 7 (4 . 3010324 636510516540 1451263010636) 7 × 7 8 11 (9 . 8864321 414815928 A 8313 2426) 11 × 11 8 p L p (4 , χ π ω − 3 p , Q ) 2 (1 . 1010100 011101010110 0110000110001001011001001001001010011001010011101) 2 × 2 7 3 (1 . 0220110 000101201002 201220101112220112012211122211111 ) 3 × 3 0 5 (4 . 2412400 332233002334 0322023204112410) 5 × 5 0 7 (1 . 3463113612416 1206563156313232) 7 × 7 0 11 (6 . 4840181 608604237370 43861) 11 × 11 0 T able 10. Splitting ﬁeld of x 4 − x 3 + 3 x 2 − 2 x + 4, where C ( χ π ) = 5 · 41. n = 2 e (2 , M π ) − 1 = 2 − 3 · 5 · 41 p R 2 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0010010 110101000111 1110110001011010111110011100101011001110011011000) 2 × 2 6 3 (2 . 0021220 111220210202 122211101102111121010022202210012111112202) 3 × 3 4 5 (4 . 2413230 212134320403 404223440242144143010) 5 × 5 4 7 (5 . 6622665 100264660251 241453165220553) 7 × 7 4 11 (2 . 0 A 3806825 36 A 50856633 A 596 3 AA ) 11 × 11 4 p L p (2 , χ π ω − 1 p , Q ) 2 (1 . 1000110 010001001011 0110100011001000110001101100111111011111110000000) 2 × 2 5 3 (1 . 0000112 110000221101 100101100220112122121100200111220 ) 3 × 3 0 5 (3 . 3132232 420011032344 4242424141204000) 5 × 5 1 7 (6 . 2064221123141 0105634103112315) 7 × 7 0 11 ( 4 . 8488929 AA 248277 415458863) 11 × 11 0 n = 4 e (4 , M π ) − 1 = 2 − 5 · 3 2 · 5 3 · 41 3 p R 4 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0110101 101101100100 1110001011011001001100001000100001111001011100010) 2 × 2 9 3 (2 . 1220122 100211112211 1110002102122122100002111212000210102220) 3 × 3 10 5 (3 . 0343321 324121423210 2131324212010224400) 5 × 5 8 7 (3 . 2134142 453160546510 4510422214051) 7 × 7 8 11 ( A. 180567 AA 0545 A 59652909 A 250) 11 × 11 8 p L p (4 , χ π ω − 3 p , Q ) 2 (1 . 1110000 111110010111 0011000101000001100001100100001000011000001000111) 2 × 2 6 3 (1 . 2110210 100202111000 001112120121020120200000202111102 ) 3 × 3 0 5 (1 . 2241420 131214213013 1112334430123200) 5 × 5 1 7 (6 . 6620060521664 3655434600606423) 7 × 7 0 11 (3 . 5059939560 A 2 AA 911068 44 A 7) 11 × 11 0 42 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT T able 11. Splitting ﬁeld of x 4 − 2 x 3 + 6 x 2 − 5 x + 2, where C ( χ π ) = 13 · 17. n = 2 e (2 , M π ) − 1 = 2 − 3 · 13 · 17 p R 2 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 1110011 101101111011 0000110110011010010001100010000011011000000100011) 2 × 2 7 3 (1 . 0011101 101010222210 1020210100122002111011122111210020001 0011) 3 × 3 5 5 (4 . 12002342224442 011223100142021013114003331) 5 × 5 4 7 (6 . 0065051 533250322312 551556340446112) 7 × 7 4 11 (9 . 5860238 2412 A 681650341 722097) 11 × 11 4 p L p (2 , χ π ω − 1 p , Q ) 2 (1 . 0101000 100000000000 1000000101011111111001100001000001100111111100100) 2 × 2 6 3 (2 . 0212122 200002021202 200211122100200001001012201202000 ) 3 × 3 1 5 (2 . 1230000 402311044214 1103112001044200) 5 × 5 0 7 (5 . 6503564405546 6446205106662353) 7 × 7 0 11 (6 . 38803 A 213 738 A 731776470 A 4 ) 11 × 11 0 n = 4 e (4 , M π ) − 1 = − 2 − 5 · 3 2 · 13 3 · 17 3 p R 4 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0101011 001010010011 0100011110011010001011010001010001001110111101010) 2 × 2 10 3 (1 . 1111010 110110112222 1120020012111221020112000222100222201 10) 3 × 3 11 5 (2 . 3000410 004401344130 02331411231121430312) 5 × 5 8 7 (2 . 6324624 136046311531 5231424221502) 7 × 7 8 11 ( A. 492999533 29 A 1125214 A 939 A 7) 11 × 11 8 p L p (4 , χ π ω − 3 p , Q ) 2 (1 . 0010011 101010000111 0011000011111110110110001001100110010111000011110) 2 × 2 7 3 (1 . 2001110 201210120022 112022021221001220010010202222220 ) 3 × 3 1 5 (4 . 3111103 423122330322 4440300102101300) 5 × 5 0 7 (3 . 2325064501066 4211233114156026) 7 × 7 0 11 (6 . 722 A 82921 527466612856 704) 11 × 11 0 T able 12. Splitting ﬁeld of x 4 + 3 x 2 − 6 x + 6, where C ( χ π ) = 2 2 · 3 2 · 11. n = 2 e (2 , M π ) − 1 = − 2 − 1 · 3 2 · 11 p R 2 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0001111 111001101011 1101100110001100000010001000010010011001000110100) 2 × 2 6 3 (2 . 0011210 002222111210 0210220110110110021120200000112000120 1101) 3 × 3 2 5 (4 . 32211042044110 431112434230011130322040334) 5 × 5 4 7 (6 . 4600034 066060161654 33631101363300) 7 × 7 5 11 (4 . 355421 A 54 525023347874 A 49229) 11 × 11 3 p L p (2 , χ π ω − 1 p , Q ) 2 (1 . 1110100 010010100001 1100111111110011101011000001101111010111100101100) 2 × 2 5 3 (1 . 2022010 222110011100 120101011011011020010020011221012 ) 3 × 3 0 5 (3 . 2201202 020113301422 3242323022213400) 5 × 5 0 7 (2 . 4331451155620 6561022451602200) 7 × 7 1 11 (7 . 5148145 37963976728 A 46 001) 11 × 11 0 n = 4 e (4 , M π ) − 1 = − 2 · 3 8 · 11 3 p R 4 ,p ( M π ) /D ( M π ) 1 / 2 ,p 2 (1 . 0001110 100100000011 1110111000101001111110011011101001101001001110101) 2 × 2 11 3 ( 2 . 022201012000 1012000101112011222120000210000112021000101001) 3 × 3 8 5 (4 . 2000241 003224110043 23111441041141020122) 5 × 5 8 7 (5 . 1253644 025610330036 4331512021150) 7 × 7 8 11 (6 . 6966231 827671 A 91984 A 43 4 A 7) 11 × 11 7 p L p (4 , χ π ω − 3 p , Q ) 2 (1 . 1011100 010110101010 0110110100100101011000111100000010010100110001101) 2 × 2 6 3 (1 . 0221110 122220100011 212020012222021100000010022110011 ) 3 × 3 0 5 (3 . 4420010 444313234102 1214030231112200) 5 × 5 0 7 (4 . 4400354050022 6200364534021346) 7 × 7 0 11 (5 . 7665842 90652809 A 28501 199) 11 × 11 0 ON THE p -ADIC BEILINS ON CONJECTURE FOR NUMBER FIELDS 43 T able 13. Splitting ﬁeld of x 6 − x 5 − 6 x 4 + 7 x 3 + 4 x 2 − 5 x + 1, where C ( χ π ⊗ i d) = 2 2 · 5 · 19. n = 3 e (3 , M E π ) − 1 = 3 − 2 · 19 2 p R 3 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 01010101110 0101010) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 111100111 000011101) 2 × 2 2 ) 3 1 ⊗ ((2 . 12112212121 0110121) 3 × 3 4 ) + ζ 3 ⊗ ((1 . 100202121 120121202) 3 × 3 4 ) 5 1 ⊗ ((4 . 40202000124 1122012) 5 × 5 3 ) + ζ 3 ⊗ ((1 . 402102204 113114001) 5 × 5 4 ) 7 1 ⊗ ((2 . 14041543264 22162) 7 × 7 6 ) + ζ 3 ⊗ ((5 . 144436263 0564332) 7 × 7 6 ) 11 1 ⊗ ((6 . 9026829 596822) 11 × 11 6 ) + ζ 3 ⊗ ((8 . 048680429 402 A ) 11 × 11 6 ) p L p (3 , χ π ⊗ ω − 2 p , Q ) 2 1 ⊗ ((1 . 01011101010 1101000) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 111111110 001001011) 2 × 2 2 ) 3 1 ⊗ ((1 . 12001210102 2112222) 3 × 3 0 ) + ζ 3 ⊗ ((2 . 212202101 110022110) 3 × 3 0 ) 5 1 ⊗ ((1 . 22232100442 0021231) 5 × 5 0 ) + ζ 3 ⊗ ((1 . 123432442 320044212) 5 × 5 0 ) 7 1 ⊗ ((6 . 21564455355 1135401) 7 × 7 0 ) + ζ 3 ⊗ ((1 . 642236344 041330514) 7 × 7 0 ) 11 1 ⊗ ((6 . 6 A 0036761 338492361) 11 × 11 0 ) + ζ 3 ⊗ ((8 . 749011 AA 36945 40145) 11 × 11 0 ) n = 5 e (5 , M E π ) − 1 = 2 7 · 3 2 · 5 2 · 19 4 · (37754 + 43673 ζ 3 ) p R 5 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 00001110000 1001101) 2 × 2 9 ) + ζ 3 ⊗ ((1 . 101010111 001110110) 2 × 2 9 ) 3 1 ⊗ ((1 . 11011012001 0021222) 3 × 3 12 ) + ζ 3 ⊗ ((2 . 020220021 122220210) 3 × 3 12 ) 5 1 ⊗ ((3 . 02001301424 2432200) 5 × 5 7 ) + ζ 3 ⊗ ((1 . 220134043 203242432) 5 × 5 7 ) 7 1 ⊗ ((4 . 15141551 064450) 7 × 7 10 ) + ζ 3 ⊗ ((6 . 531200602 44502) 7 × 7 10 ) 11 1 ⊗ ((4 . 285500134) 11 × 11 12 ) + ζ 3 ⊗ ((2 .A 3934821906 ) 11 × 11 10 ) p L p (5 , χ π ⊗ ω − 4 p , Q ) 2 1 ⊗ ((1 . 11001001000 1000101) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 111111100 101010110) 2 × 2 2 ) 3 1 ⊗ ((1 . 21102101010 1222221) 3 × 3 0 ) + ζ 3 ⊗ ((2 . 122212200 211101011) 3 × 3 0 ) 5 1 ⊗ ((1 . 40122240112 3304240) 5 × 5 0 ) + ζ 3 ⊗ ((1 . 432224122 043203014) 5 × 5 0 ) 7 1 ⊗ ((2 . 44144235634 2000541) 7 × 7 0 ) + ζ 3 ⊗ ((3 . 664531413 660053644) 7 × 7 0 ) 11 1 ⊗ ((6 . 3 A 4 A 10 A 600523025 18) 11 × 11 0 ) + ζ 3 ⊗ ((4 . 7 AA 8843572652 457 A 4) 11 × 11 0 ) T able 14. Splitting ﬁeld of x 6 − x 5 − 8 x 4 − x 3 + 12 x 2 + 7 x + 1, where C ( χ π ⊗ i d) = 2 2 · 5 · 31. n = 3 e (3 , M E π ) − 1 = 2 − 2 · 3 − 2 · 31 2 · ζ 3 p R 3 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 10010001011 1101000) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 111010100 101111010) 2 × 2 0 ) 3 1 ⊗ ((1 . 22122020001 1201020) 3 × 3 4 ) + ζ 3 ⊗ ((2 . 020111011 200022220) 3 × 3 4 ) 5 1 ⊗ ((1 . 14113034200 2111122) 5 × 5 3 ) + ζ 3 ⊗ ((4 . 342032011 212400441) 5 × 5 3 ) 7 1 ⊗ ((4 . 45055563555 66640) 7 × 7 6 ) + ζ 3 ⊗ ((3 . 061002366 4215014) 7 × 7 6 ) 11 1 ⊗ ((7 . 3700473 650 A 19) 11 × 11 6 ) + ζ 3 ⊗ ((1 . 3659 A 523968 80) 11 × 11 6 ) p L p (3 , χ π ⊗ ω − 2 p , Q ) 2 1 ⊗ ((1 . 10111000110 0101000) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 010100111 010111101) 2 × 2 4 ) 3 1 ⊗ ((2 . 11111222220 1000112) 3 × 3 0 ) + ζ 3 ⊗ ((1 . 120112022 212020122) 3 × 3 0 ) 5 1 ⊗ ((4 . 32114211332 1234330) 5 × 5 0 ) + ζ 3 ⊗ ((1 . 130134403 242024422) 5 × 5 0 ) 7 1 ⊗ ((5 . 53000630063 2330553) 7 × 7 0 ) + ζ 3 ⊗ ((2 . 200601212 403230506) 7 × 7 0 ) 11 1 ⊗ ((2 . 0076292 7347972431 A ) 11 × 11 0 ) + ζ 3 ⊗ (( A. 191 A 17800 51997 A 650) 11 × 11 0 ) n = 5 e (5 , M E π ) − 1 = − 2 6 · 3 3 · 5 2 · 31 4 · (2 + ζ 3 ) · (2 − ζ 3 ) · (7 − 15 ζ 3 ) · (54 + 31 ζ 3 ) p R 5 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 01110001100 0010010) 2 × 2 8 ) + ζ 3 ⊗ ((1 . 011100000 000101011) 2 × 2 8 ) 3 1 ⊗ ((2 . 20220202211 1202222) 3 × 3 14 ) + ζ 3 ⊗ ((1 . 112002110 200111001) 3 × 3 15 ) 5 1 ⊗ ((3 . 14143302414 1141322) 5 × 5 8 ) + ζ 3 ⊗ ((2 . 002014030 014311244) 5 × 5 7 ) 7 1 ⊗ ((6 . 50135302 661002) 7 × 7 10 ) + ζ 3 ⊗ ((4 . 043561145 36113) 7 × 7 10 ) 11 1 ⊗ ((5 .A 3306 A 68921) 11 × 11 10 ) + ζ 3 ⊗ ((3 . 85 A 6650344 A ) 11 × 11 10 ) p L p (5 , χ π ⊗ ω − 4 p , Q ) 2 1 ⊗ ((1 . 10011011001 1011000) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 110110100 001011101) 2 × 2 4 ) 3 1 ⊗ ((2 . 20110121121 0020202) 3 × 3 0 ) + ζ 3 ⊗ ((1 . 212021202 011010201) 3 × 3 0 ) 5 1 ⊗ ((4 . 01101301241 0044000) 5 × 5 0 ) + ζ 3 ⊗ ((1 . 431333301 302323324) 5 × 5 0 ) 7 1 ⊗ ((2 . 13055236524 3450401) 7 × 7 0 ) + ζ 3 ⊗ ((2 . 601221155 363314062) 7 × 7 0 ) 11 1 ⊗ ((9 . 53274 A 487 A 14818064) 11 × 11 0 ) + ζ 3 ⊗ ((9 . 89 A 62565082 5167861) 11 × 11 0 ) 44 AMNON BESSER, P AUL BUCKINGHAM, ROB DE JEU, AND XA VIER-FRANC ¸ OIS ROBLOT T able 15. Splitting ﬁeld of x 6 − x 5 − 8 x 4 + 5 x 3 + 19 x 2 − 4 x − 11, where C ( χ π ⊗ id) = 5 · 139. n = 3 e (3 , M E π ) − 1 = 2 − 5 · 3 − 2 · 139 2 p R 3 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 00011010011 1100100) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 000011011 101110010) 2 × 2 5 ) 3 1 ⊗ ((2 . 10221201021 2021001) 3 × 3 4 ) + ζ 3 ⊗ ((2 . 122111112 211000220) 3 × 3 4 ) 5 1 ⊗ ((1 . 40021131011 4320013) 5 × 5 3 ) + ζ 3 ⊗ ((1 . 434230430 334022423) 5 × 5 3 ) 7 1 ⊗ ((2 . 25231540031 06061) 7 × 7 6 ) + ζ 3 ⊗ ((2 . 320002126 0361110) 7 × 7 6 ) 11 1 ⊗ ((9 . 4472801 A 61 A 44) 11 × 11 6 ) + ζ 3 ⊗ ((1 . 72322 21727046) 11 × 11 6 ) p L p (3 , χ π ⊗ ω − 2 p , Q ) 2 1 ⊗ ((1 . 00010010101 0000101) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 000010100 000000110) 2 × 2 3 ) 3 1 ⊗ ((2 . 11101210122 1111220) 3 × 3 2 ) + ζ 3 ⊗ ((1 . 010011111 201220212) 3 × 3 0 ) 5 1 ⊗ ((3 . 24024331102 4013404) 5 × 5 0 ) + ζ 3 ⊗ ((4 . 142342004 244233331) 5 × 5 2 ) 7 1 ⊗ ((6 . 54334343063 1523116) 7 × 7 1 ) + ζ 3 ⊗ ((2 . 342260265 363120032) 7 × 7 0 ) 11 1 ⊗ ((4 . 8521241 A 647 A 883282) 11 × 11 0 ) + ζ 3 ⊗ ((1 . 462202203 56163464 A ) 11 × 11 0 ) n = 5 e (5 , M E π ) − 1 = 2 − 4 · 3 2 · 5 2 · 139 4 · (4 − 5 ζ 3 ) p R 5 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 00010011110 0110101) 2 × 2 10 ) + ζ 3 ⊗ ((1 . 011111001 011110010) 2 × 2 9 ) 3 1 ⊗ ((1 . 20200100211 2012221) 3 × 3 13 ) + ζ 3 ⊗ ((1 . 021221211 222021020) 3 × 3 12 ) 5 1 ⊗ ((3 . 40433204143 1011004) 5 × 5 7 ) + ζ 3 ⊗ ((3 . 213210304 101144004) 5 × 5 7 ) 7 1 ⊗ ((4 . 11031052 300040) 7 × 7 10 ) + ζ 3 ⊗ ((4 . 034114530 00015) 7 × 7 10 ) 11 1 ⊗ ((6 . 852318853 4) 11 × 11 11 ) + ζ 3 ⊗ ((5 . 2 A 39 A 447198) 11 × 11 10 ) p L p (5 , χ π ⊗ ω − 4 p , Q ) 2 1 ⊗ ((1 . 10100101100 1000000) 2 × 2 4 ) + ζ 3 ⊗ ((1 . 000100011 110101010) 2 × 2 3 ) 3 1 ⊗ ((1 . 20021201010 2010102) 3 × 3 1 ) + ζ 3 ⊗ ((1 . 122011112 111021002) 3 × 3 0 ) 5 1 ⊗ ((3 . 12411321320 4101323) 5 × 5 0 ) + ζ 3 ⊗ ((2 . 114244212 313224030) 5 × 5 3 ) 7 1 ⊗ ((6 . 43002431346 6045624) 7 × 7 0 ) + ζ 3 ⊗ ((5 . 065261416 212661354) 7 × 7 0 ) 11 1 ⊗ ((5 .A 88931 A 0106 A 0172 A 3) 11 × 11 0 ) + ζ 3 ⊗ ((3 .A 4151 A 485 571789978) 11 × 11 0 ) T able 16. Splitting ﬁeld of x 6 − x 5 − 8 x 4 + 11 x 3 + 2 x 2 − 5 x + 1, where C ( χ π ⊗ id) = 2 2 · 3 2 · 13. n = 3 e (3 , M E π ) − 1 = − 2 − 2 · 3 2 · 13 2 · ζ 3 p R 3 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 10110011010 1111100) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 001000001 010110111) 2 × 2 0 ) 3 1 ⊗ ((2 . 02122120110 0120120) 3 × 3 5 ) + ζ 3 ⊗ ((2 . 020012210 111121212) 3 × 3 7 ) 5 1 ⊗ ((3 . 00043103121 1204434) 5 × 5 6 ) + ζ 3 ⊗ ((1 . 012314421 212444244) 5 × 5 6 ) 7 1 ⊗ ((3 . 04264453632 00511) 7 × 7 6 ) + ζ 3 ⊗ ((1 . 103550461 5611621) 7 × 7 6 ) 11 1 ⊗ ((8 . 9799 A 36158285 ) 11 × 11 6 ) + ζ 3 ⊗ ((9 . 22992 3 AAA 0734) 11 × 11 6 ) p L p (3 , χ π ⊗ ω − 2 p , Q ) 2 1 ⊗ ((1 . 10010011100 1100011) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 101001000 011101110) 2 × 2 4 ) 3 1 ⊗ ((1 . 02211011020 1121122) 3 × 3 2 ) + ζ 3 ⊗ ((2 . 102110200 120211001) 3 × 3 0 ) 5 1 ⊗ ((2 . 42440144341 4003124) 5 × 5 0 ) + ζ 3 ⊗ ((4 . 440233034 214103134) 5 × 5 0 ) 7 1 ⊗ ((5 . 42102602205 5251515) 7 × 7 0 ) + ζ 3 ⊗ ((4 . 510213122 044500212) 7 × 7 0 ) 11 1 ⊗ (( A. 187435006 878255608) 11 × 11 0 ) + ζ 3 ⊗ ((6 . 47020 389393 AA 07 A 54) 11 × 11 0 ) n = 5 e (5 , M E π ) − 1 = see Example 7.6 p R 5 ,p ( M E π ) /D ( M E π ) 1 / 2 ,p 2 1 ⊗ ((1 . 00111110001 0111111) 2 × 2 7 ) + ζ 3 ⊗ ((1 . 100110001 101001011) 2 × 2 8 ) 3 1 ⊗ ((1 . 10022222000 1012001) 3 × 3 15 ) + ζ 3 ⊗ ((1 . 201100102 012122102) 3 × 3 15 ) 5 1 ⊗ ((1 . 30202143312 0040233) 5 × 5 8 ) + ζ 3 ⊗ ((1 . 414010322 010320220) 5 × 5 8 ) 7 1 ⊗ ((6 . 54022035 336014) 7 × 7 10 ) + ζ 3 ⊗ ((1 . 314341602 50212) 7 × 7 10 ) 11 1 ⊗ ((9 . 5 A 062420171) 11 × 11 10 ) + ζ 3 ⊗ (( A. 873665790 A 8) 11 × 11 10 ) p L p (5 , χ π ⊗ ω − 4 p , Q ) 2 1 ⊗ ((1 . 10111110110 1011100) 2 × 2 2 ) + ζ 3 ⊗ ((1 . 001110000 011000011) 2 × 2 4 ) 3 1 ⊗ ((2 . 00101211111 0022002) 3 × 3 1 ) + ζ 3 ⊗ ((2 . 011221210 002201001) 3 × 3 0 ) 5 1 ⊗ ((3 . 24112012321 1444141) 5 × 5 1 ) + ζ 3 ⊗ ((2 . 103430110 032401010) 5 × 5 0 ) 7 1 ⊗ ((5 . 45411606026 2152231) 7 × 7 0 ) + ζ 3 ⊗ ((5 . 160242506 230053303) 7 × 7 0 ) 11 1 ⊗ ((9 . 9950088486 A 64 79846) 11 × 11 0 ) + ζ 3 ⊗ (( A. 17849206188 8054088) 11 × 11 0 )

On the p-adic Beilinson conjecture for number fields

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