Milnor $K$-group attached to a torus and Birch-Tate conjecture

MILNOR K -GR OUP A TT A CHED TO A TORU S AND BIR CH-T A TE CONJE C TURE T AKAO Y AMAZAK I T o Pr ofessor T atsuo Kim ur a on the o c c asion of his sixtieth birthday Abstra ct. W e formulate (and prov e und er a certain assumption) a conjecture relating the order of Somek aw a’s Milnor K -group attac hed to a torus T and the v alue of the Artin L -function attac hed to the co character group of T (regarded as an A rtin represen tation) at s = − 1. The case T = G m reduces to the classical Birch-T ate conjecture. 1. Introduction The Birc h-T ate conjecture states that, for a totally real num b er field K , the follo w ing equalit y should hold: | ζ K ( − 1) | = | K 2 ( O K ) | | W 2 ( K ) | . Here ζ K ( s ) is th e Dedekind zeta fun ction of K , K 2 ( O K ) is the second K -group of the ring O K of int egers in K , and W 2 ( K ) = H 0 ( K, Q / Z (2)) . This equalit y is pr ov ed up to a p o we r of 2 by Wi les [21] . W e shall f ormulate a conjecture with a co efficien ts in a torus T , whic h r educes to the Birc h-T ate conjecture recalled ab ov e when T = G m . Let T b e a torus o v er a num b er field K , and let X ( T ) = Hom( G m , T ) b e the co characte r group. Then X ⊗ C is a finite dimensional representa tion of the a bsolute Galois group G K of K with fi nite image, i.e. an Artin represen tation. Let L K ( X ( T ) , s ) b e the Artin L -fun ction attac hed to it. W e set W T ( K ) = H 0 ( K, X ( T ) ⊗ Q / Z (2)). W e write K T ( K ) f or the Milnor K -group K ( K ; T , G m ) attac h ed to T and G m in tro duced b y Somek a w a [17 ]. (W e will recall the d efinition of K T ( K ) in § 2.) In § 4, w e define a subgroup K T ( O K ) of K T ( K ). W hen T = G m , w e hav e iden tities L K ( X ( T ) , s ) = ζ K ( s ) , W T ( K ) ∼ = W 2 ( K ) , K T ( O K ) ∼ = K 2 ( O K ) . W e prop ose the f ollo w ing generalization of the Birc h-T ate conjecture, w h ic h we shall p ro v e for a certain class of to ri. Date : Nov ember 21, 2018. 1991 Mathematics Subje ct Classific ation. Primary: 11R70, Secondary: 11R42, 19F15. Key wor ds and phr ases. Birc h-T ate conjecture, A rtin L -fun ction, Milnor K -groups attac hed to tori. 1 Conjecture 1.1. L et T b e a torus over a total ly r e al numb er field K . Assume T is split by a total ly r e al field. Then the e quality | L K ( X ( T ) , − 1) | = | K T ( O K ) | | W T ( K ) | should hold. Remark 1.2. The assumption that T is split b y a totally real n umber field implies that L K ( X ( T ) , − 1) is a non-zero (ratio nal) num b er (cf. pro of of Theorem 4.8). In § 2, we introd uce a condition for a torus (o v er an arbitrary field) to ‘admit a moti vic in terpretation’, and p r o v e the follo wing. Prop osition 1.3. A torus split by a meta-cyclic extension admits a motivic interpr e tation. (A finite Galois e xtension E /F of fields is c al le d meta-cyclic if al l Sylow sub gr oups of Gal( E /F ) ar e cyclic.) W e also ha v e some examples of tori whic h admits a motivic in terpretation without b eing split b y a meta-cyc lic extension (see Remark 2.10). Our main r esu lt is the follo wing. Theorem 1.4. L et L/K b e an extension o f to tal ly r e al fields, and let T b e a torus over K split by L . If T admits a motivic interpr etation, then the e quality | L K ( X ( T ) , − 1) | = | K T ( O K ) | | W T ( K ) | holds up to a p ower of 2 . This r esult will be pro v ed in § 4, where we also pro ve an analo gous result for a toru s o v er a global field of p ositiv e c haracteristic. In § 3, we study K T ( k ) for a torus T o v er a lo cal field k . 1.1. Con v entions . F or a field F , w e fi x an algebraic closure ¯ F , and all algebraic extension of F is supp osed to b e a subfield of ¯ F . W e w rite G F for the absolute Galois group of F . F or a torus T o v er a F , w e wr ite X ( T ) = Hom( T , G m ) for the co charac ter group of T . Let A b e an ab elian group. F or a non-ze ro in teger n , we wr ite A [ n ] and A/n f or th e k ernel and cok ernel of the map n : A → A. W e d efi ne A T or = ∪ n A [ n ] (resp . A div = Im(Hom( Q , A ) → Ho m( Z , A ) = A )) to b e the subgroup of t orsion elemen ts (r esp. the maximal divisible subgroup) in A . F or a prime num b er p , we d efine A [ p ∞ ] = ∪ n A [ p n ] (resp. A p − div = Im(Hom( Z [ 1 p ] , A ) → Hom ( Z , A ) = A )) to b e the su b group of p -primary torsion elemen ts (resp . the maximal p -divisible subgrou p ) in A . W e wr ite A/ div = A/ A div and A/p − div = A/ A p − div . When a group G acts on A , we write A G and A G for the in v arian ts and coin v ariants of A b y G . 2. Milnor K -group a tt ached to a torus In this section, F will b e an arbitrary fi eld. 2 2.1. Definition and basic prop erties. S omek a wa [17] has in tro du ced the Milnor K - group K ( F ; G 1 , . . . , G r ) at tac hed to a family of semi-a b elian v arieties G 1 , . . . , G r o v er F . In this pap er, w e only need a sp ecial case where G 1 = T is a torus, G 2 = G m and r = 2. T o ease the notation, we p u t K T ( F ) = K ( F ; T , G m ) . It is defined a s a quotie nt (1) K T ( F ) = M E /F T ( E ) ⊗ E ∗ /R, where E ru ns all fi nite extensions of F , and R is the group generated b y the eleme nts of the follo wing form: • (Pro jection f orm ula) L et E 1 /E 2 /F b e a to we r of fin ite extensions, and let a ∈ T ( E 1 ) , b ∈ E ∗ 2 . T h en N E 1 E 2 ( a ) ⊗ b − a ⊗ R E 1 E 2 ( b ) is a generator of R . Here N E 1 E 2 : T ( E 1 ) → T ( E 2 ) and R E 1 E 2 : E ∗ 2 ֒ → E ∗ 1 are the norm and restricti on maps r esp ectiv ely . • (W eil recipro cit y) Let F ( C ) b e a fun ction field of one v ariable ov er F , and let S b e the set of all normalized d iscr ete v aluation on F ( C ) o v er F . F or v ∈ S , we write O v (resp. F v ) for the v aluation rin g (resp . the residue field). Let a ∈ T ( F ( C )) and b, c ∈ F ( C ) ∗ . Set S ( b ) = { v ∈ S | v ( b ) 6 = 0 } . Assume that a ∈ T ( O v i ) if v 6∈ S ( b ). Then X v ∈ S ( b ) a ( v ) ⊗ ∂ v ( b, c ) + X v ∈ S \ S ( b ) ∂ v ( a, c ) ⊗ b ( v ) is a generator of R . Here ∂ v is the lo cal symb ol defined in [15], w hile a ( v ) ∈ T ( F v ) and b ( v ) ∈ F ∗ v denote the the redu ction of a and b resp ectiv ely . (Recall that ∂ v ( b, c ) is the usual ta me sym b ol.) The class of a ⊗ b ∈ T ( E ) ⊗ E ∗ in K T ( F ) is w ritten b y { a, b } E /F . W e recall some prop erties of this group. Lemma 2.1 ([17] Th eorem 1.4.) . The c orr esp ondenc e { a, b } E /F 7→ N E F { a, b } defines a c anonic al isomorphism K G m ( F ) ∼ = K 2 ( F ) , wher e the right hand side is the usu al se c ond K -gr oup, and N E F is the norm map. W e often identify K G m ( F ) with K 2 ( F ) by this isomorphism. Lemma 2.2 ([17] Prop osition 1.5) . L et T b e a torus over F , and let n b e a natur al numb er invertible in F . Then we have a homomor phism h T F : K T ( F ) /n → H 2 ( F , T [ n ] ⊗ µ n ) 3 c al le d the Galois symb ol. This map satisfies that h T F ( { a, b } E /F ) = Cor E F (( a ) ∪ ( b )) for any finite extension E /F , a ∈ T ( E ) a nd b ∈ E ∗ . Her e ( a ) ∈ H 1 ( E , T [ n ]) denotes the image of a by the c onne cting homomorphism asso ciate d to the exact se que nc e 1 → T [ n ] → T n → T → 1 , and similarly f or ( b ) ∈ H 1 ( E , µ n ) . Remark 2.3. By th e Merkurjev-Suslin theorem [9], the Galois sym b ol h T F ab o v e is bijec- tiv e when T = G m . It is conjectured by Somek a w a [17] that h T F should b e alwa ys inj ective . In Prop osition 2.11 b elo w, w e shall sh o w the injectivit y o f h T F under a certain assump tion on T . Ho w ev er, th e sur j ectivit y do es n ot hold in general. (F or example, see Prop osition 3.4). See [16] for a rela ted result. Lemma 2.4 ([16] Lemma 3) . L et E /F b e a finite sep ar able extension of fields. L et T b e a torus over E , and let S = Res E F T b e the Weil r estriction. Then, we have a n isomorp hism K T ( E ) ∼ = K S ( F ) . A sequ ence of algebraic group s G ′ → G → G ′′ o v er F is called Zariski exact if G ′ ( E ) → G ( E ) → G ′′ ( E ) is exact for any extension E /F . Lemma 2.5 ([16] Lemma 2) . L et R → S → T → 0 b e a Zariski exact se quenc e of tor i over F . Then the se quenc e K R ( F ) → K S ( F ) → K T ( F ) → 0 is exact as wel l. 2.2. Motivic in terpretat ion. W e recall Lic hten baum’s w eigh t t w o motivic complex. Review 2.6. Let Z (2) b e the we ight t w o m otivic complex, wh ic h is a t w o-term complex of discrete G F -mo dules (concen trated on degrees on e and t wo ) constructed by Lic h tenbaum [6, 7 ]. W e recall some prop erties of Z (2). (1) There is a canonical isomorphism H 2 ( F , Z (2)) ∼ = K 2 ( F ). (2) W e h a v e H 3 ( F , Z (2)) = 0. (3) If n ∈ Z is inv ertible in F , then we ha ve a triangle Z (2) n → Z (2) → µ ⊗ 2 n → Z (2)[1]. (4) If the c haracteristic p of F p ositiv e, t hen we ha v e a triangle Z (2) p s → Z (2) → ν s (2)[ − 2] → Z (2)[1], where ν s (2) is the seco nd logarithmic Hodge-Witt sheaf of lev el s . (5) There is a pro d uct map Z (1) ⊗ L Z (1) → Z (2). (Here Z (1) ∼ = G m [ − 1].) Let T b e a torus o ve r F , and let X = X ( T ) b e the co c haracter group of T . F or a fi nite extension E /F , w e hav e a homomorphism (2) T ( E ) ⊗ E ∗ ∼ = H 1 ( E , X ⊗ Z (1)) ⊗ H 1 ( E , Z (1)) ∪ → H 2 ( E , X ⊗ Z (2)) N E F → H 2 ( F , X ⊗ Z (2)) deduced b y the pro duct and norm maps . 4 Definition 2.7. W e say T admits a motivic in terpretation if the homomorphism (2) induces, via eq. (1 ), an isomorp hism K T ( F ) ∼ = → H 2 ( F , X ⊗ Z (2)) . W e exp ects an y torus admits a motivic in terpretation. In the next subs ection, we prov e this u nder a certain assu m ption. Remark 2.8. (1) I t follo ws from Lemma 2.4 a nd Sh ap ir o’s lemma that, if a toru s T o v er F admits a motivic in terpretation, then the base c hange T ⊗ F E of T by a finite sep arable extension E /F admits a motivic in terpretation as we ll. (2) In o rd er to pro v e that the map (2) fact ors through K T ( F ), one has to sho w that it kills R in eq. (1). There is no difficulty in pro ving this f or the pro jection form ula. As for the W eil reciprocity , it seems that a natural wa y to p ro v e this is to use the w eigh t three motivic complex Z (3) (and t o sh o w the v anishing of H 3 ( F ( C ) , X ⊗ Z (3)) → ⊕ v H 2 ( F ( v ) , X ⊗ Z (2)) sum → H 2 ( F , X ⊗ Z (2)) where F ( C ) is the function field of an irred u cible s mo oth prop er curv e C o ve r F , and v r uns all closed p oint s of C ). If one used V o ev o dsky’s defi n ition of Z ( r ), th is w ould follo w from the Gysin sequence [22]. Ho w ev er, V o ev o dsky’s theory is dev eloped und er the assumption of the resolution of singularit y . Because we will also consider the global fields of p ositive c haracteristic, we a v oid th e use of V o evodsky’s theory . See also [10] for a related r esult. 2.3. T ori split by a meta-cyclic extension. W e recall some facts from [3 ]. A torus P o v er F is c alled quasi-trivia l if P is isomorph ic to ⊕ i Res E i F G m , wh ere E i runs a family of finite tensions of F . A torus Q ov er F is called flasqu e if H 1 ( E , X ( Q )) = 0 for all fi n ite extension E /F . A toru s I o ver F is called inv ertible if there exists a to ru s I ′ o v er F suc h that I ⊕ I ′ is qu asi-trivial. W e ha ve implica tions ‘quasi-trivial ⇒ inv ertible ⇒ flasque’. If T is a torus o v er F split by E , then th er e exists an exact sequence (3) 0 → Q → P → T → 0 , where P (resp. Q ) is a quasi-trivial (resp . flasque) torus o ve r F sp lit b y E . W e call (3) a flasque resolutio n of T . A flasque r esolution (3) is un iqu e up to a direct summand of a quasi-trivial torus in P and Q . Prop osition 2.9. L et T b e a torus over F , and let (3) b e a flasque r esolution of T . If Q is invertible, then T admits a motivic interpr etation. Pr o of. Review 2.6 (1) and Lemma 2.1 sh o w that a split torus admits a motivic interpreta- tion. By Lemma 2.4 and Shapiro’s lemma, the same holds for a qu asi-trivial torus, hence also for an in v ertible torus. Assume a torus T admits a flasqu e resolution (3). If Q is inv ertible, then H 1 ( F ′ , Q ) = H 3 ( F ′ , X ( Q ) ⊗ Z (2)) = 0 for an y extension F ′ /F b y Hilb er t 90 and Review 2.6 (2). T his 5 in p articular im p lies th at (3) is Z ariski exact, and w e hav e by Lemma 2.5 a comm utativ e diagram with exact ro ws K Q ( F ) → K P ( F ) → K T ( F ) → 0 ↓ ∼ = ↓ ∼ = ↓ H 2 ( F , X ( Q ) ⊗ Z (2)) → H 2 ( F , X ( P ) ⊗ Z (2)) → H 2 ( F , X ( T ) ⊗ Z (2)) → 0 , sho wing the wel l-defined ness and bijectivit y of the righ t v ertical map.  Pr o of of Pr op osition 1.3 . It follo w s fr om a result of End o-Miya ta [4] (see also [3]) that a flasque to rus split b y a meta-c yclic extension is alwa ys inv ertible. No w Prop osition 1.3 is a c onsequence of Proposition 2.9.  Remark 2.10. W e giv e a few examples of a torus T whic h sat isfies the assumption of Prop osition 2.9 without being split by a meta-c yclic extension. (1) Let E /F b e a finite Galois extension whic h is not meta-cyclic. Let T b e the k ernel of the norm m ap Res E F G m → G m . T hen the dual torus ˇ T of T satisfies the assumption of Prop osition 2.9 , since it fits into an exac t sequence 0 → G m → Res E F G m → ˇ T → 0. (2) Let C b e an in tegral prop er curve ov er F whose normaliza tion is isomo rp hic to the pro jectiv e line P 1 . Assume that all singular p oin ts on C are of co ordinate axes t yp e (cf. [20]). Then the generalized Jaco bian v ariety T of C is a torus satisfying the assum ption of Prop osition 2.9. Indeed, s u c h T fits in to an exact sequence 0 → ⊕ s Res F s F G m → ⊕ s ⊕ t ∈ S ( s ) Res F t F G m → T → 0 , where s runs all s in gular p oints of C , and S ( s ) is th e inv erse image of s b y the normalization map. 2.4. A few auxiliary results. Prop osition 2.11. L et T b e a tor us over F and let X = X ( T ) . Assume that T admits a motivic interpr etation, and that n ∈ Z is invertible in F . Then we have an isomorph ism H 0 ( F , T [ n ] ⊗ µ n ) ∼ = H 1 ( F , X ⊗ Z (2))[ n ] and exact se quenc es 0 → H 1 ( F , X ⊗ Z (2)) / n → H 1 ( F , T [ n ] ⊗ µ n ) → K T ( F )[ n ] → 0 0 → K T ( F ) /n → H 2 ( F , T [ n ] ⊗ µ n ) → H 3 ( F , X ⊗ Z (2))[ n ] → 0 . Pr o of. This follo ws from the distinguish ed triangle X ⊗ Z (2) n → X ⊗ Z (2) → T [ n ] ⊗ µ n → X ⊗ Z (2)[1] deduced fr om Review 2.6 (3).  6 Corollary 2.12. L et T b e a torus over F and let X = X ( T ) . Assume T a dmits a motivic interpr etation. L et p b e a prime d iffer ent fr om the char acteristic of F . F or a natur al numb er r , we put ˆ X p ( r ) = lim ← X ⊗ µ ⊗ r p n . Then we have isomorphism s K T ( F )[ p ∞ ] / div ∼ = H 2 ( F , ˆ X p (2)) T or ∼ = H 1 ( F , X ⊗ Q p / Q p (2)) / div . (Her e H 2 ( F , ˆ X p (2)) denotes the c ontinuous Galois c oho molo gy.) Pr o of. This pro of is almost iden tical to [19] T h eorem 3.5. S et M = T [ p ] ⊗ µ p . W e ha ve a comm utativ e diagram with exact ro ws 0 → K T ( F )[ p ] → K T ( F ) p → K T ( F ) → K T ( F ) /p → 0 ↑ surj ↓ h ↓ h ↓ inj H 1 ( F , M ) → H 2 ( F , ˆ X p (2)) → H 2 ( F , ˆ X p (2)) → H 2 ( F , M ) , where the left and righ t vertica l arr o ws are th e maps in Pr op osition 2.11, and h is th e ‘con tin uous symb ol’ defin ed b y the same wa y as T ate [19]. Since H 2 ( F , ˆ X p (2)) h as no p -divisib le subgroup (cf. [19] Prop osition 2.1), we see ke r ( h ) = K T ( F ) p − div and cok er( h ) T or = 0 . This implies that K T ( F )[ p ∞ ] → H 2 ( F , ˆ X p (2)) T or is a surjection whose k ernel is K T ( F )[ p ∞ ] div = K T ( F ) div [ p ∞ ]. This p ro v es the fir st identit y . T he second iso- morphism is giv en b y [19] Prop osition 2. 3.  Lemma 2.13. L et T b e a torus over F . Then H 3 ( F , X ( T ) ⊗ Z (2)) i s a torsion gr oup of finite exp onent. Mor e over we ha ve H 3 ( F , X ( T ) ⊗ Z (2))[ p ∞ ] = 0 if cd p ( F ) ≤ 2 for a prime p 6 = Char( F ) , or if [ F : F p ] ≤ p for p = Char( F ) . Pr o of. W e set X (2) = X ( T ) ⊗ Z (2). W e tak e a finite separable extension E /F whic h splits T . Then w e know H 3 ( E , X (2)) = 0 by Review 2.6 (2). By th e norm argumen t, we see that H 3 ( F , X (2)) is an n ihilated by n = [ L : K ]. T o pro v e the second assertio n, we write n = p k m with ( p, m ) = 1 . By Review 2.6 (3) (resp. (4) ), H 3 ( F , X (2))[ p ∞ ] injects to H 3 ( F , T [ p k ] ⊗ µ p k ) (resp. H 1 ( F , X ( T ) ⊗ ν k (2))) w hen p 6 = Ch ar( F ) (resp. p = Char( F )), whic h is trivial b y assu mption.  Lemma 2.14. L et T b e a torus over a field F of p ositive char acteristic p . A ssume [ F : F p ] ≤ p . Then K T ( F ) and K T ( F )[ p ∞ ] ar e p -divisible. Pr o of. (Cf. [18] p. 205.) It is enough to sho w the p -divisibilit y of K T ( F ). W e tak e x ∈ T ( E ) , y ∈ E ∗ where E /F is a fin ite extension. Be cause the n orm maps ( E 1 /p ) ∗ → E ∗ and T ( E 1 /p ) → T ( E ) are bijectiv e, there exist x ′ ∈ T ( E 1 /p ) , y ′ ∈ ( E 1 /p ) ∗ suc h that N ( x ′ ) = x, N ( y ′ ) = y . Then w e h a v e { x, y } E /F = p { x ′ , y ′ } E 1 /p ,F , and we are done.  3. Local field When k is a local fi eld, we can pro v e that an y torus o v er k admits a motivic in terpre- tation. Th is is an immediate consequence of Theorem 1.3 if k = R (or k = C ). 7 3.1. Non-arc himedean lo cal field. Lemma 3.1. L et T b e a torus over a non-ar chime de an lo c al field k . (1) K T ( k ) is the dir e ct sum of a finite gr oup and a uniquely divisible gr oup. (2) If p = Char( k ) > 0 , then K T ( k )[ p ∞ ] = 0 . (3) L et k 1 /k b e a finite extension. Then the no rm map N k 1 k : K T ( k 1 ) → K T ( k ) is surje ctive . (4) L et m b e a natur al numb er inve rtible in k . Then, the Galois symb ol K T ( k ) /m → H 2 ( k , T [ m ] ⊗ µ m ) is bije c tiv e. (5) L et ˆ X ( r ) = lim ← X ( T ) ⊗ µ ⊗ r n and Q / Z ( r ) ′ = lim → µ ⊗ r n , wher e n runs thr ough nat ur al numb ers prime to the char acteristic of k . Then we have isomorphism s K T ( k ) T or ∼ = K T ( k ) / div ∼ = H 2 ( k , ˆ X (2)) ∼ = H 1 ( k , X ⊗ Q / Z (2) ′ ) / div ∼ = ˆ X (1) G k . Pr o of. W e tak e a fi nite Galois extension k ′ /k that splits T . It is pro v ed in [8] that K 2 ( k ′ ) is th e direct su m of a finite group and a uniquely divisible group. By the norm argument, this shows that K T ( k ) is the direct sum of a un iquely d ivisible group and a torsion group of fi nite exp onent. If Char( k ) = p > 0, then K T ( k )[ p ∞ ] is b oth divisible (by Lemma 2.14) and o f finite exp onen t, hence trivial. This prov es (2). W e p ro v e (3). When Ch ar( k ) = 0, th is is prov ed in [23] Prop osition 3.1. The same pr o of w orks as w ell when p = Char( k ) > 0, if [ k 1 : k ] is p rime to p . Th e general case can b e reduce to this case. Indeed, th e map K T ( k 1 ) div → K T ( k ) div induced by N k 1 k is surjectiv e by the n orm argumen t. Thus it suffice to sho w the surjectivit y of K T ( k 1 ) / div → K T ( k ) / div, whic h is equiv alen t to that of K T ( k 1 )[ n ] → K T ( k )[ n ], where n is the exp onent of K T ( k ) T or . By (2), w e kno w that n is prime to p . Hence we are reduced to the case [ k 1 : k ] is pr im e to p b y the norm argumen t. W e pro v e (4) and (5). W e ha v e a commutat ive diag ram K T ( k ′ ) /m ∼ = H 2 ( k ′ , T [ m ] ⊗ µ m ) ∼ = T [ m ] G k ′ ↓ N ↓ Cor k ′ k ↓ pro j. K T ( k ) /m h → H 2 ( k , T [ m ] ⊗ µ m ) ∼ = T [ m ] G k . The upp er h orizon tal map is bijectiv e b y the Me rkur jev-Suslin Theorem [9]. Th e r ight v ertical map is surjectiv e because it is ind uced by the identit y map on T [ m ]. This shows the su r jectivit y of h . This also shows that the k ernel of h ◦ N is P σ ∈ Gal ( k ′ /k ) (1 − σ ) K T ( k ′ ), whic h is killed b y N due to th e ‘pro jection formula’ relation. In view of the su rjectivit y of N pro v ed in (3), this sh o ws (4). No w (5 ) is a n immediate consequence. Lastly , w e p ro v e (1). If n is the exp onent of K T ( k ) T or , we hav e K T ( k ) T or ∼ = K T ( k ) /n ∼ = H 2 ( k , T [ n ] ⊗ µ n ) , 8 b y (4). Sin ce the right h and side is a fi nite group, w e see that K T ( k ) T or is finite. This completes the pro of.  Theorem 3.2. L et T b e a torus o ver a non-ar chime de an lo c al field k . Then T admits a motivic interpr etation. Pr o of. W e tak e a finite Galois extension k ′ /k which splits T . W e set X (2) = X ( T ) ⊗ Z (2 ) . W e are going to sh o w that (2) ind uces the homomorph ism ρ fitting in to the commutati ve diagram K T ( k ′ ) ∼ = H 2 ( k ′ , X (2)) ↓ N ↓ K T ( k ) ρ → H 2 ( k , X (2)) . The righ t ve rtical map is surjectiv e. Indeed, setting T ′ = ker[Res k ′ k T → T ], we h a v e a distinguished triangle X ( T ′ ) ⊗ Z (2) → Res k ′ k X (2) → X (2) → X ( T ′ ) ⊗ Z (2)[1] , but we ha v e H 3 ( k , X ( T ′ ) ⊗ Z (2)) = 0 by Lemma 2 .13. The left v ertical map N is also surjectiv e by Lemma 3.1 (3). Lemma 3.1 (5) shows that the k ernel of N is generated by the elemen ts of the form x − σ ( x ) with x ∈ K T ( k ′ ) and σ ∈ Gal( k ′ /k ). S u c h an elemen t is kille d in H 2 ( k , X ⊗ Z (2)) as wel l. This sho w the existe nce and su rjectivit y of ρ . Since K T ( k ) div is uniqu ely divisible, one see s that ρ | K T ( k ) div is injectiv e by the norm argumen t. On the ot her hand, ρ | K T ( k ) T or is also injectiv e as the comp osition K T ( k ) T or ∼ = K T ( k ) /n → H 2 ( k , X (2)) /n → H 2 ( k , T [ n ] ⊗ µ n ) (here n is the exp onen t of K T ( k ) T or ) is b ijectiv e b y Lemma 3.1 (4). No w the theorem follo ws from Lemma 3.1 (1).  Remark 3.3. If p is a prime different from the residue c haracteristic of k , then H 1 ( k , X ⊗ Q p / Z p (2)) div = 0 . If f u rther T has go o d reduction T v o v er the residue field F , then H 1 ( k , X ⊗ Q p / Z p (2)) ∼ = T v ( F )[ p ∞ ]. 3.2. Arc himedean lo cal field. Because K 2 ( C ) is uniquely d ivisible, K T ( C ) is uniquely divisible for an y torus T ov er C . Any to rus T o v er R admits a moti vic in terpretation by Theorem 1.3 . W e see that K T ( R ) is the direct sum of th e finite group K T ( R ) T or and the uniquely divisible group K T ( R ) div . W e need to kno w the structure o f K T ( R ) T or . Note that any torus o v er R is isomorphic to a direct sum of copies of tori app earing in the follo wing prop osition. Prop osition 3.4. We have K T ( R ) T or ∼ = Z / 2 Z (r esp. 0 ) if T = G m (r esp. if T = Res C R G m or k er[Res C R G m → G m ] ). Mor e over, for any even natur al numb er n , the exact se qu enc e 0 → K T ( R ) /n → H 2 ( R , T [ n ] ⊗ µ n ) → H 3 ( R , X ( T ) ⊗ Z (2)) 9 is isomorphic to the fol lowing se quenc e: 0 → Z / 2 Z → Z / 2 Z → 0 if T = G m 0 → 0 → 0 → 0 if T = Res C R G m 0 → 0 → Z / 2 Z → Z / 2 Z if T = k er [Res C R G m → G m ] . Pr o of. The case T = G m is we ll-kno wn. The other ca ses can b e deduced from Lemma 2.4 and the exact s equ ence 1 → k er[Res C R G m → G m ] → Res C R G m → G m → 1.  4. Global fiel d Let K b e a global field. F or a place v of K , w e write K v for the completion of K with resp ect to v . F or a finite place v of K , w e write F v (resp. K nr v ) for the residue field of v (resp. the maximal unr amified extension of K v ). When K is a n umb er field, we write O K for the ring of integ ers in K , and set C = Sp ec( O K ). When K is of p ositive charac teristic, w e assu me K i s the function field of a smooth pro j ectiv e irred u cible curve C o ve r a fin ite field F . 4.1. Blo c h-Mo ore exact sequence. W e r ecall some kno wn results. Theorem 4.1. (1) (So mekawa [17 ] ) L et T b e a torus over K . Set X = X ( T ) and ˆ X ( r ) = lim ← X ⊗ µ ⊗ r n wher e n runs thr ough natur al numb ers invertible in K . L et m b e the or der of the finite gr oup ˆ X (1) G K . Then we have an exact se q u enc e K T ( K ) → ( M v 6|∞ ˆ X (1) G K v ) ⊕ ( M v |∞ K T ( K v ) /m ) → ˆ X (1) G K → 0 . (2) (Mo or e [11] , Garland [5] ) Ther e exists an exact se quenc e 0 → W K 2 ( K ) → K 2 ( K ) → ⊕ v : non c omplex µ ( K v ) → µ ( K ) → 0 , wher e W K 2 ( K ) is a finite gr oup (so-c al le d wild kernel). In p articular, K 2 ( K ) is a torsion gr oup without any p -divisible sub gr oup for any p rime p . In the nu mb er field c ase, this also implies the finiteness of K 2 ( C ) = k er[ K 2 ( K ) → ⊕ v 6|∞ F ∗ v ] . (In the function field c ase, we have W K 2 ( K ) = K 2 ( C ) .) W e shall pro ve the fin iteness of the k ernel of the first map in (1) w hen T admits a motivic in terpretation in Prop osition 4 .6 b elo w. 10 4.2. Definition of K T ( O K ) and K T ( C ) . Let T b e a torus o v er K , and let X = X ( T ). By Theorem 4.1 and the norm argument , we see that K T ( K ) is a torsion group without an y p -divisib le su bgroup for an y p rime p . Hence, by Lemma 2.14 w e hav e K T ( K )[ p ∞ ] = 0 if p = Char( K ) > 0. Definition 4.2. F or eac h prime p 6 = Char( K ), w e define K T ( C )[ p ∞ ] = k er [ K T ( K )[ p ∞ ] → Y v 6| p ∞ H 1 ( K nr v , X ⊗ Q p / Z p (2)) G F v ] . Here the v -comp onent of the map is the composition K T ( K )[ p ∞ ] → K T ( K v )[ p ∞ ] ∼ = H 1 ( K v , X ⊗ Q p / Z p (2)) → H 1 ( K nr v , X ⊗ Q p / Z p (2)) G F v , where the second isomo rp hism is giv en b y T heorem 3.2 and Remark 3.3. W e then define K T ( C ) = ⊕ p 6 =Char( K ) K T ( C )[ p ∞ ] . In the n umb er field case, w e al so write K T ( C ) = K T ( O K ). Remark 4.3. Let p b e a prime different from Char( K ). (1) When T admits a motivic in terpretation, w e ha v e an isomorphism K T ( K )[ p ∞ ] ∼ = H 1 ( K, X ⊗ Q p / Z p (2)) / div b y Corollary 2.12. The corank of H 1 ( K, X ⊗ Q p / Z p (2)) is r 2 dim T , where r 2 is the n umb er of complex places on K b y [19 ] C orollary to Theorem 6.5. Hence, if fur ther K is totally real or of p ositiv e c h aracteristic, then w e ha v e K T ( K )[ p ∞ ] ∼ = H 1 ( K, X ⊗ Q p / Z p (2)) . (2) When T has goo d reduction at a fi nite place v , w e ha v e H 1 ( K v , X ⊗ Q p / Z p (2)) ∼ = T v ( F v )[ p ∞ ] (cf Remark 3.3). The map K T ( K )[ p ∞ ] → T v ( F v )[ p ∞ ] can b e in terpreted by the analogous w a y as the Hilbert sym b ol (cf. [17] § 3) . In particular, w e see K 2 ( C ) = K T ( C ) if T = G m . (3) Summarizing, if K is totally real or o f p ositiv e charac teristic, and if T admits a motivic in terpretation, th en K T ( O K )[ p ∞ ] is isomorphic to the k ern el o f H 1 ( K, X ⊗ Q p / Z p (2)) → ⊕ v 6∈ S, v 6| p T v ( F v )[ p ∞ ] ⊕ v ∈ S, v 6| p H 1 ( K nr v , X ⊗ Q p / Z p (2)) G F v , where S is a finite set of places of K includin g all infinite places a nd all places where T has bad reduction. 11 4.3. Hasse principle and t he finit eness of K T ( C ) . In Prop osition 4.6 b elo w, w e pro v e the finiteness of K T ( C ) when T is a torus which admits a motivic in terpretation. In the pro of, w e need the follo wing result. Prop osition 4.4 (Hasse prin ciple) . L et T b e torus over K , and let X (2) = X ( T ) ⊗ Z (2) . (1) F or al l i ≥ 3 , we have an isomorphism H i ( K, X (2)) ∼ = ⊕ v |∞ H i ( K v , X (2)) . (2) Supp ose that T admits a motivic interpr etation. L et L/K b e a finite Sep ar able extension. F or e ach infinite plac e v of K , we cho ose a plac e w ( v ) of L ab ove v . Then we have an isomorphism of finite gr oups K T ( K ) / N L K K T ( L ) ∼ = ⊕ v |∞ K T ( K v ) / N L w ( v ) K v K T ( L w ( v ) ) . (When Char( K ) > 0 , b oth statements me an that the left hand sides ar e trivial.) Remark 4.5. It is p ossible to compute the finite group app earing in (2) explicitly b y using Pr op osition 3.4. Wh en T = G m , Prop osition 4.4 (2) is pro ve d in [1, 2]. See also [23] Prop osition 4.1 for a related result. Pr o of. Firstly , w e claim that H i ( K, X (2)) is a to rsion group of finite e xp onen t for all i ≥ 3. This is r educed to the case T = G m b y the norm argumen t. By Review 2.6 (2), we ha ve H 3 ( K, Z (2)) = 0. W e also see H 4 ( K, Z (2)) = H 3 ( K, Q / Z (2)) is a torsion group of exp onent at most 2. F or i ≥ 5, the cl aim follo ws from the sp ectral sequence E m,n 2 = H m ( K, H n ( Z (2))) ⇒ H m + n ( K, Z (2)), together with th e fact th at H n ( Z (2)) = 0 unless n = 1 , 2. Let n i b e the exp onent of H i ( K, X (2)) for i ≥ 3. W e set n to b e the prim e to Char( K )- part of n i n i +1 and put M = T [ n ] ⊗ µ n . The d istinguished triangle X (2) n → X (2) → M → X (2)[1] indu ces a commutativ e d iagram with exact r o ws: 0 → H i ( K, X (2)) → H i ( K, M ) → H i +1 ( K, X (2)) → 0 ↓ f i ↓ ∼ = ↓ f i +1 0 → ⊕ v |∞ H i ( K v , X (2)) → ⊕ v |∞ H i ( K, M ) → ⊕ v |∞ H i +1 ( K, X (2)) → 0 Here the midd le v ertical map is an isomorph ism by th e Poi tou-T ate theorem (cf. [14] § 6.3 Th ´ eor ` eme B). This s h o ws the injectivit y of f i for all i ≥ 3 and (using the injectivit y of f 4 th us obtained) the surjectivit y of f i for all i ≥ 3 as w ell. When Char( K ) = p > 0, a similar argument u s ing R eview 2.6 (4) s ho ws that H i ( K, X (2))[ p ∞ ] = 0 for all i ≥ 3. T his completes the pro of of (1). W e p ro v e (2). Let S b e the k ernel of th e norm map Res L K T → T . W e set Y = X ( S ) and Y (2) = Y ⊗ Z (2). Then we ha v e a distinguished triangle Y (2) → Res L K X (2) → X (2) → 12 Y (2)[1]. By the assumption that T admits a motivic inte rpr etation, this ind uces the exact sequence at the upp er ro w in the follo w ing comm utativ e diagram 0 → K T ( K ) /K T ( L ) → H 3 ( K, Y (2)) → H 3 ( L, X (2)) ↓ ↓ ↓ 0 → ⊕ v |∞ N L w ( v ) K v K T ( L w ( v ) ) → ⊕ v |∞ H 3 ( K v , Y (2)) → ⊕ v |∞ ⊕ w | v H 3 ( L w , X (2)) The exact sequence at at the low er ro w in the diagram is d educed in a similar w a y , b y n oting the follo wing facts ( v is a p lace of K ): (i) The base c hange of Res L K T to K v is isomorp hic to ⊕ w | v Res L w K v ( T ⊗ K K v ). (ii) Th e image of the norm map N L w K v : K T ( L w ) → K T ( K v ) is the same for all w o ver v (b ecause L/K is separable). (iii) When v is a fin ite place, the norm map N L w K v : K T ( L w ) → K T ( K v ) is s u rjectiv e b y Lemma 3.1 (3). No w the assertion follo ws since the middle and righ t v ertical map s are b ijectiv e b y (1).  Prop osition 4.6. If T is a torus over K which admits a motivic interpr etation, then K T ( C ) is a finite gr oup. Pr o of. W e tak e a finite Galois extension L/K whic h splits T . Let G = Gal( L/K ) . F or a prime p 6 = C har( K ), w e ha ve a commutati ve diagram with exact rows K T ( C L )[ p ∞ ] G → K T ( L )[ p ∞ ] G → ⊕ v 6| p ∞ [ ⊕ w | v H 1 ( L nr w , X ⊗ Q p / Z p (2)) G F w ] G → 0 ↓ ↓ f ↓ 0 → K T ( C )[ p ∞ ] → K T ( K )[ p ∞ ] → ⊕ v 6| p ∞ H 1 ( K nr v , X ⊗ Q p / Z p (2)) G F v ] → 0 . If T has go o d r eduction at v , then the v -comp onen t of the r igh t ve rtical map is an iso- morphism since it is isomorphic to ( ⊕ w | v ˆ X p (1) G L w ) G ∼ = ˆ X p (1) G K v . Hence th e k ernel of the righ t vertica l m ap is finite. By Theorem 4.1, K T ( C L ) is a finite group. By Prop osition 4.4 (2), the co ke rnel of f is a finite g roup which is trivial if p 6 = 2 or Ch ar( K ) > 0. This completes the pro of.  4.4. Isogen y. W e wr ite Q / Z (2) ′ = lim → µ ⊗ 2 n where n ru ns n atural num b ers prime to Char( K ). F or a torus T o ver K , we set W T ( K ) = H 0 ( K, X ( T ) ⊗ Q p / Z p (2)). Prop osition 4.7. L et T 1 , T 2 b e tor i over K admitting a motivic i nterpr etation. We as- sume T 1 and T 2 ar e iso genous. If Char K > 0 , then the e quality | K T 1 ( C ) | | W T 1 ( K ) | = | K T 2 ( C ) | | W T 2 ( K ) | holds. When K is a numb er field, the same e quality ho lds up to a p ower of 2 , if T 1 and T 2 ar e split by a total ly r e al field. 13 Pr o of. W e only pro v e the n um b er fi eld ca se. (Th e function field case is easier.) Let S be a finite set of places of K includin g all infinite places and all places where T 1 or T 2 ha v e bad reduction. W e fi x an o dd prime p , an d set M i = X ( T i ) ⊗ Q p / Z p (2) for i = 1 , 2. W e consider a comm utativ e diagram with exact ro ws for i = 1 , 2 0 → K T i ( O K )[ p ∞ ] → H 1 ( K, M i ) → ⊕ v 6| p H 1 ( K nr v , M i ) G F v → 0 ↓ k ↓ 0 → H 1 ( O K [ 1 pS ] , M i ) → H 1 ( K, M i ) → ⊕ v 6| p S T i,v ( F v )[ p ∞ ] → 0 , where the lo w er ro w is the lo calizati on sequence of the etale cohomology , and the sur- jectivit y of the upp er right horizon tal map is due to T heorem 4.1 (1), w hic h implies the surjectivit y of the lo w er righ t horizo ntal map as well. Thus w e ha v e an exact sequence 0 → K T i ( O K )[ p ∞ ] → H 1 ( O K [ 1 pS ] , M i ) → ⊕ v 6| p ,v | S H 1 ( K nr v , M i ) G F v → 0 . The lo caliza tion sequence also implies that W T i ( K ) = H 0 ( O K [ 1 pS ] , M i ), H 2 ( O K [ 1 pS ] , M i ) = H 2 ( K, M i ) = 0. (Th e last group is trivial b ecause H 2 ( K, M i ) is a torsion group of finite exp onent (b y the norm argument ) and H 2 ( K, M ) is p -divisible for an y p -primary torsion divisible group M ). No w w e are reduced to sho wing the follo wing equalities: ( i ) | H 1 ( O K [ 1 pS ] , M 1 ) | | H 0 ( O K [ 1 pS ] , M 1 ) | = | H 1 ( O K [ 1 pS ] , M 2 ) | | H 0 ( O K [ 1 pS ] , M 2 ) | , ( ii ) | H 1 ( K nr v , M 1 ) G F v | = | H 1 ( K nr v , M 2 ) G F v | . The iso gen y f : T 1 → T 2 implies an exact sequen ce 0 → C ⊗ µ n → M 1 → M 2 → 0 , where C = ke r( f ) and n is the p -p o w er part of th e order of C. Then w e h a v e an exact sequence 0 → ke r( a ) → H 1 ( K nr v , M 1 ) a → H 1 ( K nr v , M 2 ) → 0 , in which ker( a ) is a quotien t of a finite group H 1 ( F v , C ⊗ µ n ). Since H 1 ( F v , H 1 ( K nr v , M i )) ∼ = H 2 ( K v , M i ) = K T ( K v ) ⊗ Q p / Z p = 0, w e get an exact sequence 0 → ke r( a ) G F v → H 1 ( K nr v , M 1 ) G F v → H 1 ( K nr v , M 2 ) G F v → H 1 ( F v , ker( a )) → 0 . Since ker( a ) is a fi nite G F v -mo dule, ( ii ) follo ws. In order to pr o v e ( i ), it suffice to sho w | H 0 ( O K [ 1 pS ] , C ⊗ µ n ) | · | H 2 ( O K [ 1 pS ] , C ⊗ µ n ) | | H 1 ( O K [ 1 pS ] , C ⊗ µ n ) | = 1 . By [1 2] Theorem 8 .6.14, the left hand side is equal to Y v |∞ | H 0 ( K v , C ⊗ µ n ) | || n || v . 14 By the assumption th at b oth T 1 and T 2 are split b y a tota lly real field, C ⊗ µ n is isomorphic to a direct sum of copies of Z /n Z as G K v -mo dules for an y v |∞ . This completes the pro of.  4.5. Main result. W e no w fin ish the pr o of of our m ain result Theorem 1.4. W e reca ll the stat ement, includ ing the function fi eld case. Theorem 4.8. L e t K b e a glob al field, and let T b e a torus over T . Assume that T admits a motivic interpr etation. (1) Supp ose that K is a total ly r e al numb er field, and that T is split by a total ly r e al field L over K . Then the e quality | L K ( X ( T ) , − 1) | = | K T ( O K ) | | W T ( K ) | holds up to a p ower of 2 . (2) Supp ose Char( K ) = p > 0 . Then the e quality | L K ( X ( T ) , − 1) | = | K T ( C ) | | W T ( K ) | holds. (Bot h sides ar e r ational numb ers prime to p .) Pr o of. Since L K ( X ( T ) , s ) is real analytic (as X ( T ) b eing an in tegral rep r esen tation), it is enough to s h o w the equalit y after taking the m -th p ow er for some m ∈ Z > 0 . Both sides of the equation is stable u nder isogeny (by Prop osition 4.7). By [13], there exist tori P , Q o v er K s u c h that • Bo th P and Q are qu asi-trivial an d split by L/K . • T ⊕ m ⊕ P is isogenous to Q for some m ∈ Z > 0 . Hence w e are redu ced to the case T = Res M K G m for a su b extension M /K of L/K . 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