A rigid analytical regulator for the K_2 of Mumford curves

A RIGID ANAL YTICAL REGULA TOR F OR THE K 2 OF MUMF ORD CUR VES Ambrus P ´ al April 25, 200 9. A b st r a ct . W e construct a rigid analytical regulator for the K 2 of Mumford curves, a non-arc himedean analogue of the complex analytical Beilinson-Blo ch-Deligne regula- tor. 1. Introduction Motiv ation 1. 1. Suitably normalized, the Beilinso n-Blo ch-Deligne regulator is a morphism of functor s (for deﬁnition and pr op erties see [7], pa ges 18-23): {·} : H 2 M ( U, Z (2)) → H 1 an ( U, C ∗ ) = Hom( π ab 1 ( U ) , C ∗ ) , deﬁned on the ca tegory of Riemann sur faces such that ( i ) if U = Y − { x } , where Y is a Riema nn surface, x ∈ Y is a p oint, and f , g ∈ H 0 ( U, O ∗ ) are mer o morphic at x , then the v alue of { f ⊗ g } o n the po sitively o riented lo op a round x is the tame symbol { f , g } x at x , ( ii ) If f n → f and g n → g conv erge uniformly on compact sets, where f n , f , g n and g a re elements of H 0 ( U, O ∗ ), then the v alue of { f n ⊗ g n } on every closed lo op in U conv erges to the v alue of { f ⊗ g } on that lo op. The second prop erty follows from the integral repr esentation of the monodro my . W e w is h to co nstruct a r egulator which is analog ous to the the mo no dromy of the Beilinson-Blo ch-Deligne regulator o n the bounda ry comp onents of Riemann sur - faces with b oundary in the rigid a nalytical context. If we want the tw o prop erties ab ov e to hold, then we should deﬁne the r egulator of tw o nowhere v anishing holo- morphic functions f and g on a connected rational s ubdo main of the pro jective line around a complemen tary disk b y approximating f a nd g by rational functions and set the r egulator { f , g } as the limit of the pr o duct of tame sym b ols at p oints inside the disk. This is exactly what we will do. As an application we will construct a rigid a nalytical regula tor for the K 2 of Mumfor d curves whose prop erties str ongly resemble those of the complex analytica l Beilinson- B lo ch-Deligne regulator . It is a homomorphism: {·} : H 2 M ( X, Z (2)) → H (Γ( X 0 ) , C ∗ ) , where C is an algebraica lly closed ﬁeld co mplete with respec t to an ultrametric absolute v alue, X is a Mumford curve over C and H (Γ( X 0 ) , C ∗ ) denotes the group 2000 Mathematics Subje ct Classiﬁc ation. 19C20 (primary), 11G20 (secondary) . Ty p eset by A M S -T E X 1 2 AMBRUS P ´ AL of C ∗ -v alued har monic co chains on the oriented incidence gra ph o f the spec ia l ﬁb er X 0 of some semi-stable model X o f X over the v aluation ring of C . The pro blem of ﬁnding rigid a nalytic ana lo gues of the Beilinso n-Blo ch-Deligne reg ulator has alre a dy bee n studied for exa mple in [3 ] by Coleman. Coleman’s metho ds are ra dically diﬀerent from ours, and they cov er diﬀer e nt ground. On the o ther ha nd in the t wo pap ers [8 ] and [9] Kato c onstructed a regulato r for higher lo c a l ﬁe lds whic h is essentially the same a s our s . Partially to gener alize his r esults to ﬁelds whose v aluation is not discrete, partially to b e self-co n tained, we dev elop the foundatio ns of this theory indep endently of his w ork . Con ten ts 1.2 . In the s econd chapter we deﬁne the reg ulator ﬁrst for connected rational sub domains of the pro jective line by a pproximating holomo r phic functions on the domain by rational functions and tak ing the limit of the pro duct o f the tame symbols inside a complementary op en disk a s we alrea dy mentioned ab ov e. The main result of this section is Theorem 2.2 which gives a complete c hara cterization of our r egulator . In the third chapter we ﬁrst extend the deﬁnition of the rigid analytical reg ulator for the K 2 of connected rational sub doma ins of the pro jective line (Theor e m 3.2). Then we pr ove the inv ariance theorem (Theo rem 3 .1 1) whic h is the functorialit y prope r ty o f the regulato r fo r such sub domains. In order to do so we develop an elementary homology theory for connected r ational sub domains o f the pro jective line (Theorems 3.6 a nd 3.8). The fourth chapter is somewhat technical: it contains tw o auxilia ry results used in the next chapter. In the ﬁfth chapter we rea p the fruits of o ur lab our s when we deﬁne the rigid analytica l regulator for Mumford curves. T he la tter takes v alues in the gro up of R ∗ -v alued ha r monic co chains on the oriented incidence gr aph of the sp ecial ﬁb er of a semi-s table mo del of the Mumford curve where R is the ﬁeld o f deﬁnition. W e als o s how a recipro city law relating the v aluatio n of the rigid analytica l regulator to the (generalize d) tame sym b ol along the sp ecial ﬁber . W e a lso formulate a functoria lity pr op erty for this r egulator phrased in terms of measures on the ends of the universal cov ering of the oriented incidence gra ph of the s pe c ial ﬁber (Pro p o sition 5.6). In the last chapter we lo ok at the particula r case of the Drinfeld upp er half pla ne and we express the tame symbol at the cusps as a non- archimedean integral of the rigid a nalytic regulator (Theorem 6.5). Notation 1 . 3. In this pa pe r we will use the somewhat incorr ect notation K 2 ( X ) to denote H 2 M ( X, Z (2)) for v arious t yp es of spaces X a s the latter is r ather awkward. Ac kno wledgement 1.4. I wish to thank the CRM and the IH ´ ES, wher e this article was written, for their warm hospita lit y and the pleasant environmen t they created for pro ductiv e resea r ch. 2. The rigid anal ytical regula tor f or connected ra tional subdomains Notation 2.1. Let C b e an algebraic a lly clos ed ﬁeld co mplete with res pe c t to a n ultrametric a bsolute v alue which will b e denoted by | · | . Let | C | denote the set of v alues of the latter . Let P 1 denote the pro jective line ov er C . F or a n y x ∈ P 1 and any tw o r ational non-zer o functions f , g ∈ C (( t )) o n the pr o jective line let { f , g } x denote the tame s ymbol of the pa ir ( f , g ) a t x . W e call a set D ⊂ P 1 an op en disk if it is the ima g e of the s et { z ∈ C || z | < 1 } under a M¨ obius transfor ma tion. Recall that a subs e t U of P 1 is a connected ra tio nal sub domain, if it is non-empty and A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 3 it is the complemen t of the unio n of ﬁnitely many pair-wise disjoint op en disks. Let ∂ U deno te the set o f these co mplemen tary o pe n disks. The elements of ∂ U a r e called the bo undary comp onents of U , by slight abuse of language. L e t O ( U ), R ( U ) denote the algebr a o f ho lomorphic functions on U and the subalge bra of restrictions of rational functions, resp ectively . Let O ∗ ( U ), R ∗ ( U ) denote the group of inv ertible elements of these a lgebras. The gro up R ∗ ( U ) consists of rational functions whic h do not have p oles or zero s lying in U . F or each f ∈ O ( U ) le t k f k denote sup z ∈ U | f ( z ) | . This v a lue is ﬁnite, and makes O ( U ) a Ba nach alge br a ov e r C . W e say that the sequence f n ∈ O ( U ) conv erges to f ∈ O ( U ), denoted b y f n → f , if f n conv erges to f with respe c t to the topo logy of this Banach alg ebra, i.e. lim n →∞ k f − f n k = 0. F or every real n umber 0 < ǫ < 1 we deﬁne the sets O ǫ ( U ) = { f ∈ O ( U ) |k 1 − f k ≤ ǫ } , and U ǫ = { z ∈ C || 1 − z | ≤ ǫ } . The main result of this section is the following Theorem 2.2. Ther e is a u nique map {· , ·} D : O ∗ ( U ) × O ∗ ( U ) → C ∗ for every D ∈ ∂ U , c al le d the rigid analytic r e gulator, with t he fol lowing pr op erties: (i) F or any two f , g ∈ R ∗ ( U ) their r e gulator is: { f , g } D = Y x ∈ D { f , g } x , (ii) the r e gulator {· , ·} D is biline ar in b oth variables, (iii) the r e gulator {· , ·} D is alternating: { f , g } D · { g , f } D = 1 , (iv) if f , 1 − f ∈ O ( U ) ∗ , then { f , 1 − f } D is 1 , (v) for e ach f ∈ O ǫ ( U ) and g ∈ O ∗ ( U ) we have { f , g } D ∈ U ǫ . Remark 2.3. It is an immediate cons equence of prop erty ( v ) that the rigid analytic regulator is contin uous with resp ect to the supre mum to p o logy . Explicitly , if f a nd g a re e lement s of O ∗ ( U ), D ∈ ∂ U is a b oundar y co mpo nent, and f n ∈ O ∗ ( U ), g n ∈ O ∗ ( U ) are sequences such that f n → f and g n → g , then the limit lim n →∞ { f n , g n } D , exists, and it is equal to { f , g } D . W eil’ s Recipro cit y La w 2.4 . L et f , g b e two non-zer o r ational functions on P 1 deﬁne d over the ﬁeld C . Then the pr o duct of al l tame symb ols of the p air ( f , g ) is e qual t o 1: Y x ∈ P 1 { f , g } x = 1 . Pro of. See [12], Pr op osition 6, pages 44-4 6 . Althoug h it holds for smo o th pro jec- tive algebr a ic curves in general, we will o nly use this result in the ca s e when the curve is P 1 , when ther e is a simple direct pr o of as follows. The multiplicativ e gr oup of the function ﬁeld o f P 1 is g enerated by the elements c ∈ C ∗ , and z − a , a ∈ C . Since the tame symbols are bilinea r and a lternating, we only hav e to chec k the ident ity in the claim for pairs of these elements. This reduces o ur pr o of to three 4 AMBRUS P ´ AL cases: ( z − a, z − b ), when a 6 = b , ( z − a, z − a ) and ( c, z − a ). W e compute: Y x ∈ P 1 { z − a, z − b } x = { z − a, z − b } a { z − a, z − b } b { z − a, z − b } ∞ =( a − b ) − 1 ( b − a )( − 1) = 1 , Y x ∈ P 1 { z − a, z − a } x = { z − a, z − a } a { z − a, z − a } ∞ = ( − 1)( − 1) = 1 , Y x ∈ P 1 { c, z − a } x = { c, z − a } a { c, z − a } ∞ = cc − 1 = 1 .  Deﬁnition 2.5. Let U b e a co nnected ra tio nal sub doma in of P 1 . F or any D ∈ ∂ U and for any tw o f , g ∈ R ∗ ( U ) we deﬁne the r igid analytica l regula tor { f , g } D by the formula: { f , g } D = Y x ∈ D { f , g } x ∈ C ∗ . Since the pro duct on the right ha nd side is ﬁnite, the reg ula tor is well-deﬁned. F or ea ch D ∈ ∂ U a nd f ∈ R ( U ) le t deg D ( f ) denote the num ber of zer os z of f with z ∈ D coun ted with multiplicities minu s the num b er of p oles z of f with z ∈ D counted with multiplicities. F or e very r eal num ber 0 < ǫ < 1 w e deﬁne the set R ǫ ( U ) as the intersection R ǫ = O ǫ ( U ) ∩ R ( U ). W e also deﬁne the sets O 1 ( U ) = S 0 <ǫ< 1 O ǫ ( U ) and R 1 ( U ) = S 0 <ǫ< 1 R ǫ ( U ). Lemma 2.6. ( i) The r e gulator is a biline ar map: {· , ·} D : R ∗ ( U ) ⊗ R ∗ ( U ) → C ∗ , (ii) the r e gulator {· , ·} D is alternating: { f , g } D · { g , f } D = 1 , (iii) if f , 1 − f ∈ R ∗ ( U ) , then { f , 1 − f } D is 1 . (iv) for any f ∈ R ∗ ( U ) , c ∈ C ∗ we have { c, f } D = c deg D ( f ) . Pro of. The ﬁrst three claims hold b ecause they hold for the tame symbol. Claim ( iv ) is obvious.  Lemma 2.7. The set R ǫ ( U ) is a sub gr oup of R ∗ ( U ) . Pro of. Note that R ǫ ( U ) is a subset of R ∗ ( U ), s o w e hav e to show that it is a g r oup with resp ect to m ultiplication. If f ∈ R ǫ ( U ) then | f ( z ) | = 1 for ea ch z ∈ U by the ultrametric inequality . Hence k 1 − f − 1 k = s up z ∈ U | f ( z ) | − 1 | f ( z ) − 1 | ≤ ǫ . Similarly for any f , g ∈ R ǫ ( U ) w e have k 1 − f g k ≤ sup z ∈ U max( | 1 − f ( z ) | , | f ( z ) − f ( z ) g ( z ) | ) ≤ max( sup z ∈ U | 1 − f ( z ) | , sup z ∈ U | f ( z ) | · | 1 − g ( z ) | ) ≤ ǫ .  A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 5 Deﬁnition 2.8. W e call a set D ⊂ P 1 a closed disk if it is the imag e of the set { z ∈ C || z | ≤ 1 } under a M¨ obius transformation. This terminology might b e confusing to so me as clos ed disks ar e op en and close d in the na tur al top ology o f P 1 , just like op en disks. O n the other hand closed disks are rational s ubdo mains while op en disk s are not. F or every op en disk D let D denote the unique closed disk which co ntains D and minimal with resp e ct to this prop erty . Mor e ov er let ∂ D denote the complement of D in D : this notatio n will not ca use confusion bec ause op en disks are no t rational subdomains . F or every z ∈ C and po sitive num ber ρ ∈ | C | let D ( z , ρ ), D ( z , ρ ), D ( ∞ , ρ ) and D ( ∞ , ρ ) denote the following o pe n disks and closed disks: D ( z , ρ ) = { z ∈ C || z | < ρ } , D ( z , ρ ) = { z ∈ C || z | ≤ ρ } and D ( ∞ , ρ ) = { z ∈ C || z | > ρ − 1 } ∪ {∞} , D ( ∞ , ρ ) = { z ∈ C || z | ≥ ρ − 1 } ∪ {∞} . Let C 0 , C 00 and k denote the v aluatio n ring { z ∈ C || z | ≤ 1 } of C , its maximal ideal { z ∈ C || z | < 1 } , and its residue ﬁeld C 0 / C 00 , resp ectively . Let z ∈ k denote the reduction o f e very z ∈ C 0 mo dulo the ideal C 00 . F or every ﬁnite set S ⊂ k let U ( S ) denote the set { z ∈ C 0 | z / ∈ S } . T his set is a connected rational s ubdo main of P 1 , bec ause it is the complement of the pair-wise disjoint disks D ( ∞ , 1) and D ( s, 1), where s is an element of S ⊂ C 0 , a set of representatives of the res idue classes in S . Finally let S c denote the co mplement of every subset S of P 1 . Lemma 2. 9. Assu me that 0 ∈ S . Then every f ∈ R ǫ ( U ( S )) c an b e written in the form: f ( z ) = c Y a ∈ D ( ∞ , 1) (1 − z a ) v ( a ) · Y s ∈S Y a ∈ D ( s, 1) (1 − a − s z − s ) v ( a ) , wher e c ∈ U ǫ and for e ach a ∈ C the inte ger v ( a ) is the multiplici ty of a in the divisor of f . Pro of. The ra tional function: g ( z ) = Y a ∈ D ( ∞ , 1) (1 − z a ) v ( a ) · Y s ∈S Y a ∈ D ( s, 1) (1 − a − s z − s ) v ( a ) , is in R 1 ( U ( S )), b ecause it is a pr o duct of elemen ts of R 1 ( U ( S )), which is a group, since it is the union o f a chain o f groups. The rationa l function f ( z ) /g ( z ) is also in R 1 ( U ( S )), but it has no zer os o r p oles in C − S , so it must b e equal to the function c Q s ∈S ( z − s ) n ( s ) for some c ∈ C ∗ and n ( s ) ∈ Z . Since k f /g k = 1 , w e hav e 1 = sup z ∈ U ( S )      c Y s ∈S ( z − s ) n ( s )      = | c | . Hence c ∈ C 0 , so f /g is a rationa l function with co eﬃcients on C 0 . Its r eduction mo dulo the ideal C 00 is r ( z ) = c Q s ∈ S ( z − s ) n ( s ) . Since f ( z ) / g ( z ) is in R 1 ( U ( S )), the rational function r ( z ) is identically one on k − S . Hence it is co nstant as k is algebraic ally clo sed, therefo r e f ( z ) = cg ( z ). B y the P rop osition o f I.1.3 in [4], page 7, we know that every h ( z ) ∈ O ( U ( S )) ha s a genera liz ed Laurent expansio n: h ( z ) = ∞ X n =0 a n z n + X s ∈S ∞ X n =1 b s n ( z − a ) − n , 6 AMBRUS P ´ AL and k h ( z ) k = max (max ∞ n =0 | a n | , max s ∈S (max ∞ n =1 | b s n | )). The co ns tant term a 0 in the g e ne r alized La urent expansio n of g ( z ) is 1, so the c onstant ter m o f f ( z ) is c and we ha ve | 1 − c | ≤ k 1 − f ( z ) k ≤ ǫ .  Prop ositi o n 2.10. Every f ∈ R ǫ ( U ) c an b e written in the form: f ( z ) = Y D ∈ ∂ D f D ( z ) , wher e f D ( z ) ∈ R ǫ ( D c ) for al l D ∈ ∂ U , and t hese functions ar e uniquely determine d up to a c onstant factor in U ǫ . Pro of. It is clear that the functions f D ( z ) are uniquely determined up to a factor in U ǫ , so we only hav e to show that they e xist. Assume ﬁrs t that U is of the form U ( S ) for some ﬁnite set S ⊂ k . The claim holds trivially for the do ma in U ( S ) if the set S is empty . Other wise we mig ht as sume that 0 ∈ S by a linear change of co ordinates. In this case we know that f ( z ) = f ∞ ( z ) · Y s ∈S f s ( z ) , where f ∞ ( z ) ∈ R ( D (0 , 1)) ∩ R 1 ( U ), and f s ( z ) ∈ R ( D ( ∞ , 1)) ∩ R 1 ( U ) for all s ∈ S by Lemma 2.9. Moreover f ∞ (0) ∈ U ǫ and f s ( ∞ ) ∈ U ǫ for all s ∈ S . How ever the functions f ∞ ( z ) and f s ( z ) ar e unique up to a fa c to r in U ǫ , so f ∞ ( a ) and f s ( b ) ∈ U ǫ for all a ∈ D (0 , 1) and b ∈ D ( ∞ , 1) for all s ∈ S , which ca n be seen by anticipating the automorphisms z 7→ z + a and 1 z 7→ 1 z + 1 b of U , res pec tively . Le t 1 − f ∞ ( z ) = P ∞ n =0 a n z n . There is an N ( ǫ ) ∈ N s uch tha t | a n | < ǫ if n > N ( ǫ ). Let B = { b ∈ D (0 , 1) || b | n − m 6 = | a m /a n | , ∀ n, ∀ m ≤ N ( ǫ ) } . If b ∈ B , then max n ≤ N ( ǫ ) | a n | · | b | n = | X n ≤ N ( ǫ ) a n b n | ≤ max( | 1 − f ∞ ( b ) | , | X n>N ( ǫ ) a n b n | ) ≤ ǫ. Since sup b ∈ B | b | = 1, the inequality ab ov e implies that ma x n ≤ N ( ǫ ) | a n | ≤ ǫ , so k 1 − f ∞ ( z ) k = max n ∈ N | a n | ≤ ǫ . Hence f ∞ ∈ R ǫ ( D (0 , 1)), and a similar argument shows that f s ∈ R ǫ ( D ( ∞ , 1 )) for all s ∈ S , so the claim ho lds for domains of the form U ( S ). In the genera l case w e prov e the prop osition by induction on the cardinality of ∂ U . When ∂ U is empt y the claim is obvious. Otherwise let D b e a b oundary comp onent of U . If ∂ D " U then there is a D ′ ∈ ∂ U such that ∅ 6 = ∂ D ∩ D ′ . W e claim that D ′ ⊂ ∂ D whenever D , D ′ are tw o disjoint disks such that ∅ 6 = ∂ D ∩ D ′ , and D ∪ D ′ 6 = P 1 . W e may a ssume that ∞ / ∈ D ∪ D ′ by a linea r change of co ordinates . Let a ∈ ∂ D ∩ D ′ : then D = D ( a, ρ 1 ) and D ′ = D ( a, ρ 2 ) for some ρ 1 , ρ 2 , where D deno tes the closure of D . If ρ 2 > ρ 1 , then D ⊂ D ( a, ρ 1 ) ⊂ D ( a, ρ 2 ) which is impo ssible. Therefore ρ 2 ≤ ρ 1 , s o D ′ = D ( a, ρ 2 ) ⊆ D ( a, ρ 1 ) − D = ∂ D . W e deﬁne the relation D ′ ≤ D o n the set ∂ U by the rule D ′ ⊂ ∂ D . This is clearly a partia l ordering. Let D ⊆ ∂ U be an equiv a le nce class of minimal e lement s resp ect to this ordering. This means that all D ∈ D is minimal with resp ect to this or dering, and any D ′ ∈ ∂ U is an element of D if and only if D ≤ D ′ and D ′ ≤ D for any (and hence all) D ∈ D . Cho ose a b oundar y comp onent D ∈ D and make a linear change of co ordina tes such tha t D = D (0 , 1). Then all D ′ ∈ D is of the form D ( s, 1), whe r e s ∈ S , A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 7 and the latter is a ﬁnite subs e t of C 0 . Since these disks ar e pair-wise disjoint, S injects into k with resp e ct to the reduction mo dulo the ideal C 00 . I f S denotes the image o f S with resp ect to this ma p, then the set E = T D ∈D ∂ D is equal to U ( S ). Therefore E is an a ﬃnoid sub domain of U , and we hav e already prov ed in the previous para graph that the claim ho lds for this doma in. Hence f ( z ) = g D ( z ) · Y D ∈D f D ( z ) on this set, where g D ( z ) ∈ R ǫ ( D (0 , 1)) and f D ( z ) ∈ R ǫ ( D c ) for all D ∈ D . Since the set E is inﬁnite, the equation a bove holds for all z ∈ P 1 . Hence g D ( z ) = f ( z ) Q D ∈D f D ( z ) ∈ R ǫ ( U ) , so g D ( z ) ∈ R ǫ ( Y ), to o , where Y = U ∪ D (0 , 1). Because | ∂ Y | < | ∂ U | , the induction hypothesis implies that g D ( z ) = Q D ′ ∈ ∂ Y f D ′ ( z ) with f D ′ ( z ) ∈ R ǫ ( D ′ c ). Since ∂ U = ∂ Y ∪ D , the cla im follows.  Prop ositi o n 2.11. F or every b oundary c omp onent D ∈ ∂ U a nd for e ach f ∈ R ǫ ( U ) and g ∈ R ∗ ( U ) we have { f , g } D ∈ U ǫ . Pro of. If the cardina lit y | ∂ U | = 1 then the claim is clear, since b y W eil’s recipro city { f , g } D = 1. Hence we may assume that ∞ ∈ D c − U by choosing a n a ppropriate co ordinate function z . Since U ǫ is a g roup, it is suﬃcient to prov e the claim for g ( z ) equal to one of the following gener ators of of R ∗ ( U ): c ∈ C ∗ , z − a , where a ∈ D , and z − b , wher e b ∈ D c − U . Then { f ( z ) , c } D = Y D ′ ∈ ∂ U { f D ′ ( z ) , c } D = { f D ( z ) , c } D = c − deg D ( f D ) . But deg D ( f D ) is zero, since f D has no zeros o r p oles in D c , hence { f ( z ) , c } D = 1. Also { f ( z ) , z − a } D = { f D ( z ) , z − a } D · Y D ′ ∈ ∂ U −{ D } { f D ′ ( z ) , z − a } D = Y x ∈ D c { f D ( z ) , z − a } − 1 x · Y D ′ ∈ ∂ U −{ D } { f D ′ ( z ) , z − a } a = { f D ( z ) , z − a } − 1 ∞ · Y D ′ ∈ ∂ U −{ D } f D ′ ( a ) = f D ( ∞ ) · Y D ′ ∈ ∂ U −{ D } f D ′ ( a ) ∈ U ǫ , where we used W eil’s recipro city law in the sec o nd line. Finally { f ( z ) , z − b } D = Y D ′ ∈ ∂ U { f D ′ ( z ) , z − b } D = { f D ( z ) , z − b } D = Y x ∈ D c { f D ( z ) , z − b } − 1 x = { f D ( z ) , z − b } − 1 b · { f D ( z ) , z − b } − 1 ∞ = f D ( b ) − 1 · f D ( ∞ ) ∈ U ǫ , using again W eil’s rec ipr o city law.  8 AMBRUS P ´ AL Pro of of Theorem 2. 2. Let f and g b e e lement s of O ∗ ( U ) and let D ∈ ∂ U be a boundar y comp onent. By deﬁnitio n (see [4], page 5) there are sequences f n ∈ R ∗ ( U ), g n ∈ R ∗ ( U ) such that f n → f a nd g n → g . W e say tha t the rigid analytical reg ulator { f , g } D ∈ C ∗ exists, if the limit lim n →∞ { f n , g n } D , exists, it is a n e le men t of C ∗ , and it is indep endent of the choice of the sequences f n and g n . In this case w e de ﬁne { f , g } D to b e this limit. W e start our pr o of by showing that the r igid analytica l { f , g } D regulator exis ts. Let f n ∈ R ∗ ( U ) and g n ∈ R ∗ ( U ) b e tw o sequences such that f n → f and g n → g . W e will ﬁrst show that the sequence { f n , g n } D conv erges using Cauch y’s cr iterion. F or every 0 < ǫ < 1 there is an n ( ǫ ) ∈ N such that f n /f m ∈ R ǫ ( U ), g n /g m ∈ R ǫ ( U ) for any n , m ≥ n ( ǫ )( U ). By Pr o p osition 2.11 for any n , m ≥ n ( ǫ ): { f n , g n } D / { f m , g m } D = { f n /f m , g n } D · { f m , g n /g m } D ∈ U ǫ , so { f n , g n } D is a Cauch y sequence. If f ′ n ∈ R ∗ ( U ) and g ′ n ∈ R ∗ ( U ) are another t wo sequences s uch that f ′ n → f and g ′ n → g , then the sequences f ′′ n , g ′′ n also conv erge to f a nd g resp ectively , wher e f ′′ 2 n +1 = f n , f ′′ 2 n = f ′ n , and g ′′ 2 n +1 = g n , g ′′ 2 n = g ′ n . The limit lim n →∞ { f ′′ n , g ′′ n } D also exists, so it must be equal to the limit of its subsequences : lim n →∞ { f ′′ n , g ′′ n } D = lim n →∞ { f n , g n } D = lim n →∞ { f ′ n , g ′ n } D , hence the limit is independent of the s equences chosen. On the other hand 1 = lim n →∞ { f n , g n } D · { g n , f n } D = lim n →∞ { f n , g n } D · lim n →∞ { g n , f n } D , so this limit is non-zero , so the existence is proved. Incident ally we also proved that the r igid analytical r e g ulator sa tis ﬁe s prop erty ( iii ). Because O ∗ ( U ) is op en in O ( U ), and the inverse ma p is co ntin uous , prop er - ties ( ii ), ( iv ) and ( v ) follow fr o m Le mma 2.6 a nd Prop o s ition 2.11 , b y contin uity . Prop erty ( i ) follows by taking the sequenc e s f n = f a nd g n = g . On the other hand pr o p erties ( ii ) and ( v ) imply that any map s atisfying these prop erties m ust be contin uous in b oth v ar iables, so it is equa l to the rigid analytica l regulator if it also satisﬁes ( i ).  3. The inv ariance theorem Deﬁnition 3.1. In this c hapter w e will contin ue to use the notation of the prev ious chapter. Let U be a connected r ational sub domain of P 1 , and f , g are tw o mero- morphic functions on U . Then for all x ∈ U the functions f and g have L a urent series expans ion ar ound x , in pa r ticular their tame s ymbol { f , g } x at x is deﬁned. Let M ( U ) denote the ﬁeld of meromo r phic functions of U . F o r every x ∈ U the tame sym b ol at x extends to a homomorphism {· , ·} x : K 2 ( M ( U )) → C ∗ . W e deﬁne the gro up K 2 ( U ) as the kernel of the direct sum of tame s ymbols: M x ∈ U {· , ·} x : K 2 ( M ( U )) → M x ∈ U C ∗ . A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 9 Let k = P i f i ⊗ g i ∈ K 2 ( M ( U )), where f i , g i ∈ M ( U ), and let D ∈ ∂ U . Let moreov er Y b e a connected rationa l sub domain of U such that f i , g i ∈ O ∗ ( Y ) for all i and ∂ U ⊆ ∂ Y . Deﬁne the rig id a nalytical regulator { k } D by the formula: { k } D = Y i { f i | Y , g i | Y } D . Theorem 3.2. ( i ) F or e ach k ∈ K 2 ( M ( U )) the rigid analytic al r e gulator { k } D is wel l-deﬁne d, and it is a homomorphism {·} D : K 2 ( M ( U )) → C ∗ , ( ii ) for any two functions f , g ∈ O ∗ ( U ) we have { f ⊗ g } D = { f , g } D , ( iii ) for every k ∈ K 2 ( U ) the pr o duct of al l re gulators on t he b oun dary c omp onents of U is e qual to 1: Y D ∈ ∂ U { k } D = 1 , ( iv ) for every c onne cte d sub domain Y ⊆ U , b oundary c omp onent D ∈ ∂ Y ∩ ∂ U and k ∈ K 2 ( M ( U )) we have: { k | Y } D = { k } D . In order to prove this theor e m, w e will need three lemmas. Lemma 3.3. L et f , g ∈ O ∗ ( U ) , wher e U is a c onne cte d r ational su b domain of P 1 . Then the pr o duct of al l rigid analytic al r e gulators of the p air ( f , g ) on t he b oundary c omp onents of U is e qual to 1: Y D ∈ ∂ U { f , g } D = 1 . Pro of. Let f n , g n be t wo seque nc e s o f rational functions invertible on U which conv erge to f and g on the domain, resp ectively . Then Y D ∈ ∂ U { f , g } D = Y D ∈ ∂ U lim n →∞ { f n , g n } D = lim n →∞ Y D ∈ ∂ U Y x ∈ D { f n , g n } x = lim n →∞ Y x / ∈ U { f n , g n } x = 1 by W eil’s recipro cit y law.  Lemma 3.4. L et Y ⊆ U b e two c onne cte d r ational sub domains. Then for every f , g ∈ O ∗ ( U ) and D ∈ ∂ Y we have: { f | Y , g | Y } D = Y D ′ ∈ ∂ U D ′ ⊆ D { f , g } D ′ , wher e the empty pr o duct is understo o d to b e e qual to 1. 10 AMBRUS P ´ AL Pro of. W e may reduce immedia tely to the case when f , g ∈ R ∗ ( U ) b y approxi- mation. Then w e have: { f | Y , g | Y } D = Y x ∈ D { f | Y , g | Y } x = Y x ∈ D { f , g } x = Y D ′ ∈ ∂ U D ′ ⊆ D Y x ∈ D ′ { f , g } x ! = Y D ′ ∈ ∂ U D ′ ⊆ D { f , g } D ′ .  Lemma 3.5. Le t U b e a c onne cte d ra tional sub domain and let D ⊂ U b e an op en disk. L et f , g b e two m er omorphic functions on U which ar e holomorphic and invertible on the r ational su b domain Y = U − D . Then we have: { f | Y , g | Y } D = Y x ∈ D { f , g } x . Pro of. The mer omorphic functions f and g hav e only ﬁnitely many po les and zeros o n the disk D . Let S denote the set of these po ints. F or every such a p ole or zero s ∈ S choose a small o p e n disk D s ⊂ D cont aining s such that the pair- wise in tersectio ns o f these disks a r e empty . Let W denote the r ational s ubdo main U − ∪ s ∈ S D s . Then { f | Y , g | Y } D = Y s ∈ S { f | W , g | W } D s by Lemma 3.4 . Hence we may assume without lo osing g enerality that D = D ( a, ρ ) and a is the only p ole or zero o f f and g on D by letting W , D s play the ro le of U and D , resp ectively , in the claim above. W rite f ( z ) = ( z − a ) k f 0 ( z ), g ( z ) = ( z − a ) l g 0 ( z ), where f 0 ( z ), g 0 ( z ) are elements o f O ∗ ( U ). Let f n , g n be t wo se q uences of rational functions inv ertible on the r ational sub domain U whic h conv erge to f 0 , g 0 , resp ectively , with r esp ect to the supremum norm on U . Then { f , g } D = lim n →∞ { ( z − a ) k f n , ( z − a ) l g n } D = lim n →∞ { ( z − a ) k f n , ( z − a ) l g n } a = { ( z − a ) k , ( z − a ) l } a · lim n →∞ { ( z − a ) k , g n } a · lim n →∞ { f n , ( z − a ) l } a =( − 1) kl · lim n →∞ g n ( a ) − k · lim n →∞ f n ( a ) l = ( − 1 ) kl g 0 ( a ) − k f 0 ( a ) l = { f , g } a .  Pro of o f Theorem 3. 2. In o rder to show that the rig id analytical regulato r is well-deﬁned, we hav e to prov e that: ( a ) for e ach k ∈ K 2 ( U ) there is an aﬃnoid sub domain Y with the prop erties required in Deﬁnition 3.1, ( b ) the v alue of the r igid analytica l reg ulator is indep endent of the choice of the aﬃnoid sub domain Y , ( c ) the v alue of the r igid analytica l regula tor is indep endent of the choice of the presentation k = P i f i ⊗ g i . Let S ⊂ U a ﬁnite s et such that none of the functions f i and g i has a zero or a po le on U − S . F or ea ch s ∈ S ta ke an o p e n disk D s ⊂ U containing s such that these disks a re pair-wise disjoint. The set Y = U − S s ∈ S D s is a co nnected rational A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 11 sub do main which satisﬁes the prop erties required in Deﬁnition 3.1 , so claim ( a ) holds. Le t Y ′ be ano ther such sub domain. It is clear that the set Y ∪ Y ′ is also a connected rationa l sub domain which satisﬁes these prop erties. Lemma 3.4 applied to the inclusions Y ֒ → Y ∪ Y ′ and Y ′ ֒ → Y ∪ Y ′ implies tha t ( b ) is also tr ue. Let k = P i f ′ i ⊗ g ′ i be another pre s entation of k . By deﬁnition X i f i ⊗ g i − X i f ′ i ⊗ g ′ i = X j r j in the fr ee group g enerated by s ymbols f ⊗ g , where f , g ∈ M ( U ), and the elements r j are deﬁning rela tions of the gro up K 2 ( M ( U )), i.e. they a re of the form: f g ⊗ h − f ⊗ h − g ⊗ h , h ⊗ f g − h ⊗ f − h ⊗ g , or f ⊗ (1 − f ), f 6 = 1 . It is p ossible to choo se a connected rational sub doma in Y ⊂ U which satisﬁes the prop erties requir ed in Deﬁnition 3.1 a nd it do es not contain a ny o f the zer os or po les of the functions f i , g i , f ′ i , g ′ i , and the functions a pp ea ring in the rela tions r j . By ( ii ) and ( iv ) of Theore m 2.2 the regulato r {· , ·} D ev aluated on the relatio ns r j is equal to 1 , so the pro ducts: Y i { f i | Y , g i | Y } D = Y i { f ′ i | Y , g ′ i | Y } D are equal. Ther efore ( c ) holds, to o. The same a rgument also shows that the map {·} D : K 2 ( M ( U )) → C ∗ is a homomorphism. Claim ( ii ) is obvious, if we cho o se Y = U in the deﬁnition. W e start the pro o f of claim ( iii ) by noting that Y D ∈ ∂ Y { k | Y } D = Y D ∈ ∂ U { k } D Y D ∈ ∂ Y − ∂ U { k | Y } D by Lemma 3.4, where Y is a connected r a tional subdo main which satisﬁes the prop erties requir ed in Deﬁnition 3 .1 with respec t to some pres ent ation of k . By Lemma 3.5 the factors of the second pro duct on the right hand s ide are all equal to 1. O n the other hand Lemma 3.3 applied to Y implies that the pro duct o n the left hand s ide is equa l to 1. Hence the ﬁr st pro duct on the right hand side is equal to 1, to o. W e may as sume that k = f ⊗ g for some f , g ∈ M ( U ) by bilinearity w hile we prov e claim ( iv ). Ther e is a connected sub domain Z ⊆ U such that ∂ U ⊆ ∂ Z , the in tersection Y ∩ Z is nonempt y and f , g ∈ O ∗ ( Z ). Clearly D ∈ ∂ ( Y ∩ Z ), so it will b e suﬃcient to prov e that { f | Y ∩ Z , g | Y ∩ Z } D = { f | Z , g | Z } D . W e may even a ssume that f and g are in R ∗ ( Y ) by approximation. But in this case the cla im is obviously tr ue.  Theorem 3. 6. Ther e is a u nique set of homomorph isms deg D : M ∗ ( U ) → Z wher e U is any c onne cte d r ational sub domain and D ∈ ∂ U is a b oundary c omp onent with the fol lowing pr op ert ies: ( i ) the homomorph ism deg D is zer o on O 1 ( U ) , ( ii ) for every f ∈ R ∗ ( U ) the inte ger deg D ( f ) is the quantity deﬁne d in 2.5, ( iii ) for every f ∈ M ∗ ( U ) we have deg D ( f | Y ) = deg D ( f ) wher e Y ⊆ U is any c onne cte d r ational sub domain satisfying the pr op erty D ∈ ∂ Y . This map is equa l to the integer-v alued function introduced in [4 ], Prop ositio n I.3.1, page 1 9 , in the limited c o ntext when the latter is deﬁned. 12 AMBRUS P ´ AL Pro of. By condition ( i ) it is clear that the homomorphis m deg D restricted to O ∗ ( U ) must b e co n tinuous with r esp ect to the supremum top ology on O ∗ ( U ) and the discr ete top olog y on Z , if it exists. F o r every f ∈ M ∗ ( U ) there is a Y ⊆ U connected r ational s ub doma in such that D ∈ ∂ Y and f | Y ∈ O ∗ ( Y ), so the set of homomorphisms deg D : M ∗ ( U ) → Z should be uniq ue. Pick a n element c ∈ C with | c | > 1 and deﬁne deg D ( f ) as deg D ( f ) = log | c | ( |{ c, f } D | ) where log a is the log arithm with base a for a ny p ositive rea l num b er a . This ho - momorphism is well-deﬁned by Theorem 3.2. This function also satisﬁes condition ( ii ) of the pr o claim ab ov e b y cla im ( iv ) of Lemma 2 .3 . Hence the image of R ∗ ( Y ) with resp ect to this map lies in Z . But it is als o dense in the image o f O ∗ ( Y ) via this map, so the la tter lies in Z , too . W e may conclude that the homomorphisms deg D take integral v alues. By ( v ) o f Theorem 2 .2 the element { c, f } D is a unit in C 0 when f ∈ O 1 ( U ), so condition ( i ) is also sa tisﬁed. Condition ( i ii ) is the consequence of claim ( iv ) of Theo r em 3.2.  Deﬁnition 3.7 . F o r every U ⊂ P 1 connected rational sub doma in let Z ∂ U denote the free ab elian group with the elements of ∂ U a s free generato rs. Let H 1 ( U ) denote the quo tien t of Z ∂ U by the Z -mo dule genera ted b y P D ∈ ∂ U D . F or every D ∈ ∂ U we let D denote the class of D in H 1 ( U ) as well. Let A b denote the categ ory of ab elian gr oups. Let C r s denote the category w ho se ob jects are connected ra tional sub do mains of P 1 and whose mo rphisms a r e holomorphic ma ps b etw een them. Finally for every pair a ≤ b of num ber s in | C | let A ( a, b ) deno te the clos ed a nnulus P 1 − D (0 , a ) − D ( ∞ , 1 /b ). Of course it is a connected rationa l sub domain. Theorem 3.8. Ther e is a un ique functor H 1 : C rs → A b with the fol lowing pr op- erties: ( i ) for every U ⊂ P 1 c onne cte d r ational su b domain H 1 ( U ) is the gr oup deﬁne d in 3.7, ( ii ) for every map U → Y which is the r est riction of a pr oje ct ive line ar tr ans- formation f and D ∈ ∂ U b oundary c omp onent we have: H 1 ( f )( D ) = f ( D ) ∈ H 1 ( Y ) , ( iii ) for every f : U → D ( a, b ) holomorphic map and D ∈ ∂ U b oundary c omp o- nent we have: H 1 ( f )( D ) = deg D ( f ) D (0 , a ) ∈ H 1 ( A ( a, b )) . Pro of. W e are going to prov e ﬁrst that this functor is unique. Let h : U → Y b e a holo morphic map b etw een tw o co nnected ratio nal sub domains . W e nee d to show that H 1 ( h ) is uniquely de ter mined by the conditions ab ov e. W e may a ssume that Y has at least tw o b oundar y co mpo nents. Fix a b oundary comp onent F ∈ ∂ Y . Then for ev ery other boundar y comp onent F 6 = E ∈ ∂ Y there is a pro jective linear transformatio n j E of P 1 such that j E ( E ) = D (0 , 1 ) and ∞ ∈ j E ( F ). Then j E ◦ H maps into A ( a, b ) for some a a nd b for every F 6 = E ∈ ∂ Y . By prop erty ( ii ) and ( iii ) we have: H 1 ( h )( D ) = X F 6 = E ∈ ∂ Y deg D ( j E ◦ h ) E ∈ H 1 ( Y ) A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 13 for every b oundar y comp onent D ∈ ∂ U . In particular this cla ss is uniquely deter- mined. Let H 1 ( U ) denote the quotient H 1 ( U ) = O ∗ ( U ) / ( C ∗ O 1 ( U )) , where C ∗ ⊂ O ∗ ( U ) is the subgroup of constant functions. F or every f ∈ O ∗ ( U ) let the same letter denote its class in H 1 ( U ) as well. The degree map of Theorem 3.6 induces a bilinear pairing: deg : H 1 ( U ) × H 1 ( U ) → Z characterized b y the prop er ty: deg( f , D ) = deg D ( f ) for every f ∈ O ∗ ( U ) and D ∈ ∂ U . Prop ositi o n 3.9. The p airing deg is p erfe ct. Pro of. W e need to show the following t wo claims in order to prov e the pro p o sition: (i) for every f ∈ O ∗ ( U ) if deg D ( f ) = 0 for every D ∈ ∂ U then f ∈ C ∗ O 1 ( U ), (ii) for e very homomorphism l : H 1 ( U ) → Z there is a function f ∈ O ∗ ( U ) such that l ( h ) = deg( f , h ) for every h ∈ H 1 ( U ). W e may assume that f ∈ R ∗ ( U ) by approximation while we show the ﬁr st claim. Let P x ∈ P 1 n ( x ) x b e the divisor o f f . Let g ( z ) b e the pro duct Q D ∈ ∂ D g D ( z ), where g D ( z ) = Y x ∈ D ( z − x ) n ( x ) if ∞ / ∈ D , and g D ( z ) = Y x ∈ D ( 1 z − 1 x ) n ( x ) , otherwise. The ra tional functions f and g hav e the same divisor, s o their quotient is consta nt. Therefore it will b e eno ug h to show that g ∈ O 1 ( U ). W e will prove that the functions g D ∈ R 1 ( U ) which is suﬃcient by Lemma 2.7 . First consider the cas e when ∞ / ∈ D . In this case D = D ( c, d ) for s ome c ∈ C and d ∈ | C | . By assumption P x ∈ D n ( x ) = deg D ( f ) = 0, so the function g D ( z ) is the pro duct of factors of the for m ( z − a ) / ( z − b ), wher e a , b ∈ D . F or a ny z / ∈ D we hav e     z − a z − b     = | z − c + c − a | | z − c + c − b | = | z − c | | z − c | = 1 , so g D ∈ R 1 ( U ) as we claimed. In the other case the argument is simila r. W e may assume that ∂ U has at least tw o elements while we show the second claim. Fix a b oundary comp onent F ∈ ∂ U . As l ( F ) = − P F 6 = D ∈ ∂ U l ( D ), there is a rational function f ∈ R ∗ ( U ) suc h that deg D ( f ) = l ( D ) for ev ery other bounda r y comp o nent F 6 = D ∈ ∂ U . Clearly the function f satisﬁes the prop erty in the second claim.  The pro of of the existence of the functor H 1 is now easy: we deﬁne it a s the Z -dual of the contrav ariant functor H 1 . Condition ( i ) is automatic via the identiﬁcation betw een H 1 ( U ) and Ho m( H 1 ( U ) , Z ) furnished by P rop osition 3.9 . Prop erties ( ii ) and ( iii ) can b e veriﬁed by loo king at appropr iate test functions to co mpute the eﬀect o f H 1 ( f ). In the ﬁrst ca s e one consider s rationa l functions, in the second case the identit y function.  14 AMBRUS P ´ AL Deﬁnition 3. 10. Let U ⊂ P 1 be a connected rationa l sub domain. F o r ev ery class c ∈ H 1 ( U ) and element k ∈ K 2 ( U ) we deﬁne the regula tor { k } c as { k } c = Y D ∈ ∂ U { k } c ( D ) D , where P D ∈ ∂ U c ( D ) D is a lift of c in Z ∂ U . By claim ( iii ) of Theorem 3.2 this regulator is well-deﬁned. F or every holomorphic map h : U → Y betw een t wo connected rationa l s ubdo mains let h ∗ : K 2 ( M ( Y )) → K 2 ( M ( U )) b e the pull- back homomorphism induced by h . By restric tio n it induces a ho mo morphism K 2 ( Y ) → K 2 ( U ). Theorem 3 .11. F or any k ∈ K 2 ( Y ) and c ∈ H 1 ( U ) we have: { h ∗ ( k ) } c = { k } H 1 ( h )( c ) . Pro of. Let k = P i f i ⊗ g i ∈ K 2 ( M ( Y )), where f i , g i ∈ M ( U ). Let moreover Y ′ be a connected ra tio nal sub domain of Y such that f i , g i ∈ O ∗ ( Y ) for a ll i a nd ∂ Y ⊆ ∂ Y ′ . Ther e is a connected rational sub domain U ′ of U s uch tha t h ( U ′ ) ⊆ Y ′ and ∂ U ⊆ ∂ U ′ . The map H 1 ( U ′ ) → H 1 ( U ) induced by the inclusion is sur jective, so there is a c ′ ∈ H 1 ( Y ′ ) who s e image is c . W e c laim that { h ∗ ( k ) | U ′ } c ′ = { h ∗ ( k ) } c . W e may write c ′ as a sum c ′ = c 1 + c 2 where c 1 , c 2 can b e represented a s the linea r combination of bo undary comp onents lying in ∂ U and ∂ U ′ − ∂ U , res p ectively . W e hav e { h ∗ ( k ) | U ′ } c 1 = { h ∗ ( k ) } c by de ﬁnitio n, on the other hand { h ∗ ( k ) | U ′ } c 2 = 1 a s h ∗ ( k ) is an element of K 2 ( U ). The same argument shows that { k | Y ′ } H 1 ( h | U ′ )( c ′ ) = { k } H 1 ( h )( c ) , so it will b e suﬃcie nt to prove the claim for U ′ , Y ′ , h | U ′ , c ′ and k | Y ′ instead of U , Y , h , c and k , resp ectively . In other words we may assume that k = f ⊗ g for some f , g ∈ O ∗ ( Y ). Let f n ∈ R ∗ ( Y ) and g n ∈ R ∗ ( Y ) b e tw o sequences suc h that f n → f and g n → g . Obviously k f ◦ h − f n ◦ h k ≤ k f − f n k for all n ∈ N , so f n ◦ h → f ◦ h . The same holds for g , so { h ∗ ( f ⊗ g ) } c = lim n →∞ { h ∗ ( f n ⊗ g n ) } c and { f ⊗ g } H 1 ( h )( c ) = lim n →∞ { f n ⊗ g n } H 1 ( h )( c ) by Remark 2.3. Ther efore it is suﬃcien t to show the c laim when f , g ∈ R ∗ ( Y ). W e may also assume that ∞ / ∈ U and ∞ / ∈ Y by shrinking Y and U the same wa y as ab ov e if necessary . Lemma 3.12. F or every f ∈ O ( U ) t he fol lowing holds: for every ǫ > 0 ther e is a δ > 0 such t hat | f ( x ) − f ( y ) | < ǫ for every x , y ∈ U with | x − y | < δ . Pro of. Of cour se the claim ab ove is just the analo gue of the usual uniform conti- nu ity prop erty . The r e a son that it is no t completely obvious in this case is that C is no t lo ca lly compact. Let U ( U ) denote the set of all functions f ∈ O ( U ) whic h satisfy the pr op erty in the cla im a bove. It is clear tha t U ( U ) is a C -subalgebra of O ( U ). Mo reov er for every f ∈ O ∗ ( U ) w e have the estimate:     1 f ( x ) − 1 f ( y )     =     f ( y ) − f ( x ) f ( x ) f ( y )     ≤ k f − 1 k 2 | f ( x ) − f ( y ) | , so for every f ∈ O ∗ ( U ) ∩ U ( U ) we hav e f − 1 ∈ U ( U ), to o. Obviously z − c ∈ U ( U ) for every c ∈ C , so R ( U ) ⊆ U ( U ) by the a bove. On the other hand U ( U ) is closed A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 15 with res p ect to the supremum top olog y , so it must b e equal to the whole algebra O ( U ).  Let us re tur n to the pro of of T he o rem 3.1 1. Since ∞ / ∈ U there is a ra tional ǫ > 0 s uch that for every x ∈ U the disk D ( x, ǫ ) ⊂ U . Hence we may cho ose an inﬁnite sequence h n ∈ R ( U ) conv erging to h in the supremum top ology such that h n ( U ) ⊆ Y for all n . By the lemma ab ov e f ◦ h n → f ◦ h and g ◦ h n → g ◦ h in the supremum to p o logy . Therefor e it is suﬃcient to pr ove theorem when h ∈ R ( U ), to o. W e may also assume that c = C for some bo undary comp onent C ∈ ∂ U by linearity . Let F ∈ ∂ Y b e the unique b oundary comp onent which contains ∞ . W e may write H 1 ( h )( C ) unique ly in the fo rm: H 1 ( h )( C ) = X F 6 = D ∈ ∂ Y n ( D ) D for some n ( D ) ∈ Z . There is a clos ed annulus A ( a, b ) such that h ( z ) − c ma ps U into A ( a, b ) for every F 6 = D ∈ ∂ Y and c ∈ D . Fix a boundar y compo nent F 6 = D ∈ ∂ Y and for every c ∈ D let z ( c ) denote the num b er of zer os of the rational function h ( z ) − c lying in the op en disk C counted with multip licities. W e claim that z ( c ) is indep endent of the c hoice o f c . Firs t note that the num ber of p oles of the rationa l function h ( z ) − c lying in the o pe n disk C co unted with m ultiplicities is indep endent of the choice of c . T his num b er do es not ev en depend on D , and it will be denoted by p ( C ). By claim ( i i i ) of Theorem 3.8 w e hav e z ( c ) = deg C ( h ( z ) − c ) + p ( D ) = n ( D ) + p ( C ) which is clearly indep endent o f the choice o f c . Let z ( D ) denote this num ber . W e also claim that for e very ∞ 6 = c ∈ F the num b er z ( c ) of zeros of the r a tional function h ( z ) − c lying in the op en disk C counted with multiplicities is equal to p ( C ). W e hav e F = D ( ∞ , d ) for some rational num b er d > 0. Hence k h ( z ) k ≤ 1 / d , so we have     1 − h ( z ) − c c     =     h ( z ) c     < 1 , so deg C ( h ( z ) − c ) = 0. But z ( c ) = deg D ( h ( z ) − C ) + p ( C ) = p ( C ) by deﬁnition. F or all x ∈ P 1 let v ( x ) ∈ N denote the degree of ramiﬁcation of the ma p h : P 1 → P 1 at x . T he n { f ◦ h, g ◦ h } x = { f , g } v ( x ) h ( x ) for all x ∈ P 1 . Therefore { h ∗ ( f ⊗ g ) } C = Y x ∈ C { f ◦ h, g ◦ h } x = Y x ∈ C { f , g } v ( x ) h ( x ) = Y D ∈ ∂ Y Y y ∈ D Y x ∈ C h ( x )= y { f , g } v ( x ) y = Y F 6 = D ∈ ∂ Y Y y ∈ D { f , g } z ( D ) y · Y y ∈ F { f , g } p ( C ) y = Y F 6 = D ∈ ∂ Y Y y ∈ D { f , g } z ( D ) − p ( C ) y = { f ⊗ g } H 1 ( h )( C ) , where we used W eil’s recipro city law in the ﬁfth eq ua tion.  16 AMBRUS P ´ AL Remarks 3.13. The theorem a b ove incor p o rates tw o fundamental prop erties o f the rigid ana lytic regulator which might b e called as biho lomorphic in v ar iance and homotopy inv ariance based on the analog y explained in 1.1. The ﬁst cla ims that every biholo morphic ma p h : U → U , wher e U is any connected r a tional sub do- main of P 1 , which induces the identit y map on H 1 ( U ) leav es the rigid-a nalytic regulator inv a riant. The second cla ims that whenever we have tw o connected ra- tional sub domains Y ⊆ U and a bo undary co mpo nent D ∈ ∂ U such that there is a unique bo undary co mpo nent D ′ ∈ ∂ Y containing D then the rigid ana lytic regulator s taken at D and D ′ do no t diﬀer for elements of K 2 ( U ). Of course the bes t wa y to formu late these prop er ties is the way we did, expressing them as a func- toriality prop erty via the ho mology gr oup H 1 . The latter has a more high- brow deﬁnition us ing the ´ etale coho mology o f rigid analytic spac e s , but for our pur po ses our elementary deﬁnition was more suitable. 4. Rela tion to the generalized t ame symbol Notation 4.1. Let R b e a clos ed subﬁeld o f C : it is automatically complete with resp ect to | · | . Let U b e a c o nnected rational s ubdo main of P 1 deﬁned ov er R . This means that U = { z ∈ P 1 || f i ( z ) | ≤ 1 ( ∀ i = 1 , . . . , n ) } for some natura l num b er n and r ational functions f 1 , . . . , f n ∈ R (( t )). Let O R ( U ), R R ( U ), O ∗ R ( U ), R ∗ R ( U ) and M R ( U ) denote the alg ebra of holomorphic functions, the subalg ebra of restrictions of R -rational functions, the groups of in vertible ele- men ts of these a lg ebras and the ﬁeld of mero morphic functions on the r igid analytic space U , resp ectively . Let U denote the underlying ratio nal sub domain ov er C by slight abuse of no tation. An R -r ational boundar y component o f U is a set D ∈ ∂ U such that D is of the form D ( a, | ρ | ) or D ( ∞ , | ρ | ) for some a , ρ ∈ R . Let K 2 ( U ) R de- note the large s t subgroup o f K 2 ( M R ( U )) which maps into K 2 ( U ) under the natur a l homomorphism K 2 ( M R ( U )) → K 2 ( M C ( U )). Prop ositi o n 4.2. L et D b e an R -r ational b oundary c omp onent of U , and let k ∈ K 2 ( M R ( U )) . Then { k } D ∈ R ∗ . Pro of. W e may assume that k = f ⊗ g fo r some f , g ∈ O ∗ R ( U ) by linearity . Since R ∗ R ( U ) is dense in O ∗ R ( U ) and R is complete, we may ass ume that f and g ar e actually in R ∗ ( U ) by approximation. W e may also assume that ∞ ∈ D a fter an R -linear change of c o ordinates. Then by bilinear ity we may assume that f and g are ir r educible p olynomials in R [ t ]. Assume ﬁrst that f and g a re separ able, to o. Clearly { f , g } x can b e diﬀerent fro m 1 only if x = ∞ or x is a zero of f o r g . In the latter c a se x is an ele ment of the separ able closure R hence { f , g } x ∈ R , to o. Mor eov er if x ∈ D then x γ ∈ D , to o , where γ is any elemen t of Gal( R | R ). Also { f , g } γ x = { f , g } x γ for any x ∈ R and γ ∈ Gal( R | R ). Therefor e the pro duct Q x ∈ D { f , g } x is an element of R inv ariant under the a c tion o f Gal( R | R ), so it is in R . If f is not s eparable, then f = ( f ′ ) p n for some s eparable po lynomial f ′ whose co e ﬃcient s are in a pure ly ins eparable extensio n L o f R such that L p n ⊆ R where p is the characteristic of R . It is enoug h to s how that { f ′ , g } D ∈ L ∗ , since { f , g } D = { f ′ , g } p n D ∈ R ∗ in this c ase. The latter follows from applying the same argument to g ov er L and what we hav e just prov ed above.  A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 17 Deﬁnition 4.3. Let f , O a nd m denote the r esidue ﬁeld of R , the v aluatio n ring of R and the maximal prop er ideal of O , resp ectively . A ﬁnite subset S ⊂ k is called f -rationa l if it is the zero set of a p olynomia l with co eﬃcients in f . If S ⊂ k is f -ra tional then the set U ( S ) intro duced in Deﬁnition 2.8 is a c o nnected rational sub do main deﬁned ov er R . Lemma 4.4. Le t S ⊂ k b e an f -r ational subset. Then (i) for every f ∈ M R ( U ( S )) ∗ ther e is an f -r ational su bset S ⊆ S ′ ⊂ k such that f | U ( S ′ ) c an b e written in the form: f | U ( S ′ ) = c ( f ) f 0 , wher e c ( f ) ∈ R and f 0 ∈ O ∗ ( U ( S ′ )) with | f 0 ( z ) | = 1 for al l z ∈ U ( S ′ ) , ( ii ) the p ositive numb er | c ( f ) | do es not dep end on the choic e of S ′ or f 0 , ( iii ) the map | · | : M R ( U ( S )) ∗ → | C | given by the rule f 7→ | c ( f ) | is a non- ar chime de an absolute value on the ﬁ eld M R ( U ( S )) . Pro of. The set Z ⊂ U ( S ) of zeros and pole s of f is ﬁnite. Hence the reduction of the elements of Z with r esp ect to C 00 is a ﬁnite set, too . Since every ﬁnite subset of k is contained in a ﬁnite f -rationa l s ubset, we may as sume that f ∈ O ∗ ( U ( S )) by enlarging S if necess ary . Let g ∈ O ∗ ( U ( S )) b e another function such that k 1 − f /g k < 1. W e c laim that it is suﬃcient to pr ov e the claim ( i ) for g in order to prov e it for f . W e may as sume that g = c ( g ) g 0 where c ( g ) ∈ R and g 0 ∈ O ∗ ( U ( S )) with | g 0 ( z ) | = 1 for all z ∈ U ( S ) by e nlarging S if necess ary . W rite f as f = c ( g ) f 0 . Then k 1 − f 0 /g 0 k = k 1 − f /g k < 1 so | f 0 ( z ) | = 1 for all z ∈ U ( S ). Hence we may as sume that f ∈ R ∗ ( U ( S )) by approximation. Also note that the elements of M R ( U ( S )) ∗ satisfying cla im ( i ) for m a subgroup. Ther efore we may as s ume that f is in fact a po lynomial. Then we may write f ( z ) as: f ( z ) = c ( f ) N X n =0 a n z n with a n ∈ O and ma x N n =0 | a n | = 1. Ther e is a ﬁnite f -rational subset S ′ of k such that the reduction of the p olyno mial f 0 = P N n =0 a n z n is nowhere zero on the complement of S ′ . Clearly f 0 ∈ O ∗ ( U ( S ′ )) with | f 0 ( z ) | = 1 for all z ∈ U ( S ′ ). This prov es claim ( i ). Assume that S ⊆ S ′′ ⊂ k is another ﬁnite f -r ational subset such that f | U ( S ′′ ) = c ( f ) ′ f ′ 0 where c ( f ) ′ ∈ R and f ′ 0 ∈ O ∗ ( U ( S ′ )) with | f ′ 0 ( z ) | = 1 for all z ∈ U ( S ′′ ). O n the set U ( S ′ ∪ S ′′ ) = U ( S ′ ) ∩ U ( S ′′ ) we hav e c ( f ) f 0 = c ( f ) ′ f ′ 0 . As k f 0 k = k f ′ 0 k = 1 on this se t we mu st have | c ( f ) | = | c ( f ) ′ | as claim ( ii ) says. Cle a rly the map v is a homomorphism, so we only hav e to show that | c ( f + g ) | ≤ max( | c ( f ) | , | c ( g ) | ) for any f , g ∈ O ∗ ( U ( S )) with f + g 6 = 0 in order to pr ov e claim ( iii ). There is a n f -ra tional subset S ⊆ S ′ ⊂ k such that | c ( f ) | = k f | U ( S ′ ) k , | c ( g ) | = k g | U ( S ′ ) k and | c ( f + g ) | = k ( f + g ) | U ( S ′ ) k . The las t claim no w follows from the strong tr iangle ineq ua lity for the spectr a l norm k · k .  Deﬁnition 4.5. F or any pair S ⊆ S ′ ⊂ k of ﬁnite f -r ational subsets the inclusion U ( S ) → U ( S ′ ) induces an imbedding M R ( U ( S )) → M R ( U ( S ′ )). Under these inclusions the ﬁelds M R ( U ( S )) form a injective system: let M denote the inductive limit of this system. By part ( ii ) of the lemma ab ove the absolute v alue | · | of pa rt 18 AMBRUS P ´ AL ( iii ) is well-deﬁned o n M and makes the latter a v alued ﬁeld. The r esidue ﬁeld of M is equal to the rational function ﬁeld f ( t ) where t is the reduction o f the ident ity map z with r e sp e ct to the maxima l idea l in the v aluation ring of M . W e are going to need a mild ex tension of the tame symbol. Let F be a ﬁeld equippe d with a v aluation ν : F ∗ → Q and let O and r denote its v alua tion ring and its residue ﬁeld, resp ectively . Let mor eov er · : O → r denote the reduction mo dulo the maximal ideal of O . F or any pair of elements f , g ∈ F ∗ we ar e g oing to deﬁne their g eneralized tame symbo l T ( f ⊗ g ) ∈ r ∗ ⊗ Q as follows. Ther e is an element π ∈ F ∗ such that f = f 0 π n ( f ) and g = g 0 π n ( g ) for some integers n ( f ), n ( g ) ∈ Z and elements f 0 , g 0 ∈ O ∗ . W e let T ( f ⊗ g ) = ( − 1) ⊗ ν ( f ) ν ( g ) · f 0 ⊗ ν ( g ) · g 0 ⊗ ( − ν ( f )) ∈ r ∗ ⊗ Q . (W e only included the ﬁrst factor in the pr o duct a b ove in or der to resemble the usual formu la, but it is always equal to 1 ). One may prove the usual wa y that this symbol is well-deﬁned a nd satisﬁes the Steinberg rela tion. In particula r it induces a homomorphism T : K 2 ( F ) → r ∗ ⊗ Q which depends on the c hoice of nor malization of the v a luation v linea rly . Let ν : M ∗ → Q b e a v a luation cor resp onding the absolute v alue | · | and let T : K 2 ( M ) → f ( t ) ∗ ⊗ Q denote the co rresp onding g eneralized tame symbol. F or an y k ∈ K 2 ( M ( U )) and s ∈ S ∩ f let T s ( k ) ∈ Q denote the v alue of T ( k ) with resp ect to the unique Q - linear e xtension of the normalized v aluation at the closed p oint s ∈ P 1 f . Prop ositi o n 4.6. We have ν ( { k } D ( s, 1) ) = T s ( k ) for every k ∈ K 2 ( M R ( U ( S ))) . Pro of. The linear transfor mation z 7→ z − s maps U ( S ) biholomo rphically onto U ( ∪ x ∈ S x − s ) and interc hanges {·} D ( s, 1) and T s with {·} D (0 , 1) and T 0 , r esp ectively . Hence we may assume that s = 0. By linearity we may assume that k = f ⊗ g for s ome f , g ∈ M R ( U ( S )). Since for every f -ratio nal ﬁnite set S ⊆ S ′ ⊂ k we have { k } D (0 , 1) = { k | U ( S ′ ) } D (0 , 1) and T 0 ( k ) = T 0 ( k | U ( S ′ ) ) w e ma y assume that f = c ( f ) f 0 and g = c ( g ) g 0 where c ( f ), c ( g ) ∈ R ∗ and f 0 , g 0 ∈ O ∗ ( U ( S )) with | f 0 ( z ) | = | g 0 ( z ) | = 1 for every z ∈ U ( S ) by enlarg ing S if necessa ry . Assume tha t k 1 − f k < 1. In this case T ( f ⊗ g ) = 1 by deﬁnition and |{ f ⊗ g } D (0 , 1) | = 1 by ( v ) of Theor e m 2.2 . Hence we may assume that f and g are rational functions by approximation. W e may even as sume that f and g are poly nomials using the bilinearity of b oth sides of the equa tion we wan t to prove. In this case we may assume by multiplying c ( f ) and c ( g ) by a n element o f O ∗ , if necess ary , s uch that f 0 ( z ) = z n ( f ) (1 + f 00 ( z )) and g 0 ( z ) = z n ( g ) (1 + g 00 ( z )) , where n ( f ), n ( g ) ∈ Z and f 00 ( z ), g 00 ( z ) ∈ O [ z ] with k f 00 k < 1 a nd k g 00 k < 1. Therefore we may as sume that f 0 and g 0 are powers of z by applying the ar gument we used ab ove. In this case the claim is obvious.  5. The rigid anal ytical regula tor f or Mumf ord cur ves Deﬁnition 5.1 . Let X b e a Hausdor ﬀ topolo gical space . F or a ny R commutativ e group let M ( X , R ) denote the set of R -v a lued ﬁnitely additive measures on the A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 19 op en a nd co mpact subsets of X . When X is compa ct let M 0 ( X , R ) denote the set of measures o f total volume zero , that is the subset of those µ ∈ M ( X , R ) s uch that µ ( X ) = 0. F or ev ery ab elia n top o logical gr oup M let C 0 ( X , M ) denote the group o f compactly supp orted contin uous functions f : X → M . If M is discrete then every element of C 0 ( X , M ) is lo cally co ns tant. In this ca se for every f ∈ C 0 ( X , M ) and µ ∈ M ( X , R ) we deﬁne the mo dulus µ ( f ) of f with resp ect to µ as the Z -submodule of R generated b y the element s µ ( f − 1 ( g )), where 0 6 = g ∈ M . W e also deﬁne the int egra l o f f on X with res pec t to µ as the sum: Z X f ( x )d µ ( x ) = X g ∈ M g ⊗ µ ( f − 1 ( g )) ∈ M ⊗ µ ( f ) . Lemma 5. 2. ( a ) If f , g ∈ C 0 ( X , M ) , then f × g ∈ C 0 ( X , M × M ) , µ ( f ) , µ ( g ) and µ ( f + g ) ar e c ont aine d in µ ( f × g ) and Z X f ( x ) + g ( x )d µ ( x ) = Z X f ( x )d µ ( x ) + Z X g ( x )d µ ( x ) in M ⊗ µ ( f × g ) . ( b ) L et f ∈ C 0 ( X , M ) and m ∈ N , wher e N is also a Z -mo dule. Then m ⊗ f is in C 0 ( X , N ⊗ M ) , µ ( m ⊗ f ) ⊆ µ ( f ) and Z X m ⊗ f ( x )d µ ( x ) = m ⊗ Z X f ( x )d µ ( x ) in N ⊗ M ⊗ µ ( f ) . ( c ) L et f ∈ C 0 ( X , M ) and let φ : M → N b e a homomorphism. Then φ ◦ f is in C 0 ( X , N ) , µ ( φ ◦ f ) ⊆ µ ( f ) and Z X φ ◦ f ( x )d µ ( x ) = φ  Z X f ( x )d µ ( x )  in N ⊗ µ ( f ) . ( d ) Assume that f ∈ C 0 ( X , R ) is lo c al ly c onstant and µ ∈ M ( X , R ) is a Bor el me asure on X , which is a p ositive me asur e on t he Bor el sets of X . Then the image of the inte gr al of f on X with r esp e ct to µ under t he homomorphi sm R ⊗ µ ( f ) → R induc e d by the pr o duct is the usual L eb esgue inte gr al of f on X with r esp e ct to µ .  Deﬁnition 5.3 . Let R b e a clo sed subﬁeld of C and let K ⊂ P 1 ( R ) b e a non-empty compact subset. F or every ρ ∈ GL 2 ( R ) and z ∈ P 1 ( C ) let ρ ( z ) denote the imag e of z under the M¨ obius tra nsformation corresp onding to ρ . Let moreover D ( ρ ) denote the op en disk D ( ρ ) = { z ∈ P 1 ( C ) | 1 < | ρ − 1 ( z ) |} . Let D denote the set of op en disk s of the form D ( ρ ) where ρ ∈ GL 2 ( R ). F o r each D ∈ D let D ( K ) deno te D ∩ K . Let P ( K ) denote those subsets S of D such that the sets D , D ∈ S ar e pair-wise disjoint and the unio n o f the set D ( K ), D ∈ S form a pa rtition of K . F or e a ch S ∈ P ( K ) le t Ω( S ) denote the unique co nnected rational sub domain deﬁned ov er R with the prop erty ∂ Ω( S ) = S . Let Ω K denote the complement of K in P 1 ( C ). Then Ω K is equipp ed naturally with the structure of a rig id analytic space over R such that the op e n subsets Ω( S ), ∈ P ( K ) form a n admissible cov er by aﬃnoid sub domains. In particula r a function f : Ω K → R is holomorphic if the restriction of f onto Ω( S ) is holomor phic fo r every S ∈ P ( K ). Let O (Ω K ) and M (Ω K ) denote the R -alg ebra of holo mo rphic functions a nd the ﬁeld of mer o morphic function of Ω K , resp ectively . The latter is of cour se the quotient ﬁeld of the former . W e deﬁne K 2 (Ω K ) as the intersection of the kernels of all the tame symbols {· , ·} x inside K 2 ( M (Ω K )) where x runs thr o ugh the set Ω K . 20 AMBRUS P ´ AL Lemma 5.4. F or e ach k ∈ K 2 (Ω K ) ther e is a unique me asur e { k } ∈ M 0 ( K, R ∗ ) such that { k } ( D ( K )) = { k | Ω( S ) } D for every S ∈ P ( K ) and D ∈ S . Pro of. First we are going to show that every op en cov er U of K has a sub ordina te cov er of the form D ( K ), D ∈ S where S ∈ P ( K ). By the compa c tnes s of K ther e is a ﬁnite cov er V of the form D ( K ), D ∈ I sub ordinate to U where I ⊂ D is a ﬁnite set. W e may as sume that the union ∪ D ∈ I D is not equal to P 1 ( C ) b y re ﬁning the cover V further. Then a ny tw o disk s in I are either disjoint or equa l, hence the claim is now clear. The same argument w orks for any compact and op en subset L of K . When we apply it to the one element cover of L we get that L can b e written as the pair -wise disjoint union of s e ts of the for m D ( K ). In particula r the measure { k } is unique, if it exists. In or der to pr ove that { k } ( L ) is well-deﬁned we have to show that the pro duct Y D ∈ I { k | Ω( S ) } D ∈ R ∗ is indep endent of the choice of I and S for every S ∈ P ( K ) a nd I ⊆ S such that L = ∪ D ∈ I D ( K ). Let T ∈ P ( K ) and J ⊆ T b e ano ther pair such that L = ∪ D ∈ J D ( K ). Then there is a V ∈ P ( K ) and a M ⊆ V suc h that ∅ 6 = Ω( V ) contains Ω( S ) ∪ Ω( T ) and L = ∪ D ∈ M D ( K ). Clearly we only hav e to show that { k | Ω( U ) } D = Y E ( K ) ⊆ D ( K ) E ∈ V { k | Ω( V ) } E for every D ∈ U wher e U is either S or T . By o ur assumptions we hav e either E ( K ) ⊆ D ( K ) or E ⊂ Ω K for ev ery E ⊆ D with E ∈ V . As { k | Ω( V ) } E = 1 for disks of the la tter t ype the equa lity ab ov e fo llows from the in v a r iance theo rem. Finally note that the pro duct { k } ( K ) = Q D ∈ S { k | Ω( S ) } D is equal to one for every S ∈ P ( K ) b y ( iii ) of Theorem 3.2 hence { k } is indeed an element of M 0 ( K, R ∗ ).  Deﬁnition 5.5. Now let K and L be tw o non-empt y compact s ubsets of P 1 ( R ) and assume that a non-consta nt holomorphic map h : Ω K → Ω L of rig id analy tic spaces ov er R ar e given. Let h ∗ : K 2 ( M (Ω L )) → K 2 ( M (Ω K )) b e the pull-ba ck ho mo mor- phism induced by h . By restriction it induces a ho momorphism K 2 (Ω L ) → K 2 (Ω K ), also denoted b y h ∗ . F or every ab elian top o logical group M a nd compact Hausdo rﬀ top ological space X let e C 0 ( X , M ) denote the quotient of C 0 ( X , M ) by the g roup of M -v alued consta n t functions. The integration introduced in Deﬁnition 5 .1 in- duces a ca nonical identiﬁcation b etw een Hom( e C 0 ( X , Z ) , R ) and M 0 ( X , R ) for every R co mmu tative group when Z is discrete. W e ar e going to deﬁne a homomor phism h ∗ : e C 0 ( K, Z ) → e C 0 ( L, Z ), wher e Z is equipped with the discrete top olo g y , as follows. Given an element e f ∈ e C 0 ( K, Z ) ﬁrst cho o se one o f its representativ es f ∈ C 0 ( K, Z ). Then choose an S ∈ P ( K ) such tha t f is equal to a constant f ( D ) on the set D ( K ) for every D ∈ S . Then ther e is a T ∈ P ( L ) such that h (Ω( S )) ⊆ Ω( T ). Choos e a e g = P E ∈ T e g ( E ) ∈ Z ∂ Ω( T ) which represents H 1 ( h )( X D ∈ S f ( D ) D ) ∈ H 1 (Ω( T )) . Let g ∈ C 0 ( L, Z ) be the function given by the rule g ( z ) = e g ( E ) for every z ∈ E ( K ) and E ∈ T . W e deﬁne h ∗ ( e f ) a s the clas s of g in e C 0 ( L, Z ). One may see tha t h ∗ is A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 21 a well-deﬁned homomorphism in the same way we proved the lemma a bove. L e t h ∗ : M 0 ( L, R ) → M 0 ( K, R ) be the ho momorphism induced b y this h ∗ via the duality descr ib ed ab ov e. The following prop ositio n is a n immediate c onsequence of the deﬁnitions a nd the inv a riance theorem: Prop ositi o n 5.6. We have { h ∗ ( k ) } = h ∗ ( { k } ) ∈ M 0 ( K, R ∗ ) for every k ∈ K 2 ( M (Ω L )) .  Deﬁnition 5.7. F or any gra ph G let V ( G ) and E ( G ) denote its set of v ertices and edges, resp ectively . In this pa per we will only co nsider such or iented gra phs G which are equipp e d with an involution · : E ( G ) → E ( G ) such that fo r each edge e ∈ E ( G ) the or iginal a nd ter mina l vertices of the edge e ∈ E ( G ) are the terminal and or iginal vertices of e , res pec tively . The edge e is called the edge e with r eversed orientation. Let R be a commutativ e g roup. A function φ : E ( G ) → R is called a harmonic R - v alued co chain, if it satisﬁes the following conditions: ( i ) W e have: φ ( e ) + φ ( e ) = 0 ( ∀ e ∈ E ( G )) . ( ii ) If for an edge e we intro duce the notation o ( e ) and t ( e ) for its or iginal and terminal vertex resp ectively , then for all but ﬁnitely many e ∈ E ( G ) with o ( e ) = v we hav e φ ( e ) = 0 and X e ∈E ( G ) o ( e )= v φ ( e ) = 0 ( ∀ v ∈ V ( G )) , where by o ur assumption the sum ab ov e is well-deﬁned. W e de no te b y H ( G, R ) the gro up o f R -v alued harmonic co chains on G . Deﬁnition 5.8. A path γ on an oriented gra ph G is a seq uence of edges { e 1 , e 2 , . . . , e n , . . . } ∈ E ( G ) indexed by the set I whe r e I = N or I = { 1 , . . . , m } for some m ∈ N such that t ( e i ) = o ( e i +1 ) for every i , i + 1 ∈ I . W e say that γ is an inﬁnite path o r a ﬁnite path whether we a re in the ﬁr st or in the second case, resp ectively . W e say that a path { e 1 , . . . , e n , . . . } index e d by the set I o n an oriented graph G is without backtrac king if e i 6 = e i +1 for every for every i , i + 1 ∈ I . An o riented gra ph G is ca lle d a tree if for every pair of diﬀerent vertices v a nd w ∈ V ( G ) there is a unique ﬁnite path { e 1 , . . . , e n } without backtracking such that o ( e 1 ) = v and t ( e n ) = w . Recall that a half-line γ on an o r iented gr aph G is an inﬁnite path without backtracking. W e say that tw o ha lf-lines on an oriented graph are equiv alent if they only diﬀer in a ﬁnite graph. W e deﬁne the set Ends( G ) of ends of a tree G as the equiv alence classes of half-lines o f G . There is a natural top olog y on E nds( G ) given by the sub-basis G e , e ∈ E ( G ) where G e consists of the equiv ale nc e c la sses of half-lines of the form { e 1 , e 2 , . . . , e n , . . . } with the prop erty e 1 = e . 22 AMBRUS P ´ AL Deﬁnition 5.9. By s lightly extending the usual terminolo gy we will c all a scheme C deﬁned ov er a ﬁeld a curv e if it is r educed, lo cally of ﬁnite type and of dimension one. A cur ve C is said to have normal crossings if every singular p oint of C is an ordinary double p oint in the usual sense. F or any curve C with nor mal cros sings let e C denote its nor ma lization, and let e S ( C ) denote the pre-ima g e of the set S ( C ) of singula r p oints of C . W e deno te by Γ( C ) the o riented g raph whose set of vertices is the set of irr educible co mp onents of e C , and its se t o f edges is the set e S ( C ) such that the initial vertex o f an y edge x ∈ e S ( C ) is the irreducible comp onent of e C which contains x and the terminal vertex o f x is the irr educible comp onent which contains the unique o ther element x of e S ( C ) whic h maps with resp ect to the normalizatio n map to the same singular po in t as x . The map x 7→ x is an in volution · : E (Γ( C )) → E (Γ( C )) of the type describ ed in Deﬁnition 5 .7. Deﬁnition 5.1 0. Let O , f denote the v aluation ring of R and its r e s idue ﬁeld, resp ectively . Let U be an admissible formal s cheme o f dimension one ov er O and let U deno te the rigid ana ly tic space w e g et from U by applying Ra ynaud’s functor (for its deﬁnition see [2]). Let U 0 denote the sp ecial ﬁb er of U and assume that the curve U 0 ov er f is totally degenerate. The latter means that U 0 has normal crossing s and its irreducible co mpo nen ts are smo oth pro jective rationa l curves over f . Assume that U is biholomorphic to Ω ∂ U for some non-empty ∂ U ⊆ P 1 ( R ). In this case the g r aph Γ( U 0 ) is a tree and the topo lo gical space Ends(Γ( U 0 )) is ca nonically homeomorphic to ∂ U (see [6]). W e will use this identiﬁcation without further notice. F or every element k ∈ K 2 ( U ) let { k } deno te the function { k } : E (Γ( U 0 )) → R ∗ which is giv en by the rule { k } ( e ) = { k } (Γ( U 0 ) e ) for every edge e ∈ E (Γ( U 0 )) wher e we use the notation of Deﬁnition 5.8 and the symbol { k } on the right hand side of the equation ab ove denotes the meas ure we intro duced in Lemma 5 .4. Lemma 5.11. The fu n ction { k } lies in H (Γ( U 0 ) , R ∗ ) . Pro of. The claim is purely graph-theo retical in nature. In fact for every tree T , commutative gro up R and mea sure µ ∈ M 0 (Ends( T ) , R ) the function c ( µ ) : E ( T ) → R , given b y the rule c ( µ )( e ) = µ ( T e ) for every edge e ∈ E ( T ), is an R - v alued har monic co chain. Fix a vertex v ∈ V ( T ): then every end of T has a unique representative of the for m { e 1 , e 2 , . . . , e n , . . . } with the prop erty o ( e 1 ) = v . N ow it is clear that the sets T e , o ( e ) = v form a pa ir-wise disjoin t par tition o f Ends( T ). Therefore c ( µ ) satisﬁes prop erty ( ii ) of Deﬁnition 5.7. Similarly prop er ty ( i ) of Deﬁnition 5.7 follows from the fact that, given an edge e ∈ E ( T ), e very end of T has a unique r epresentativ e o f the form { e 1 , e 2 , . . . , e n , . . . } such that either e 1 = e or e 1 = e .  Deﬁnition 5.12. By the lemma ab ov e we hav e cons tructed a reg ulator {·} : K 2 ( U ) → H (Γ( U 0 ) , R ∗ ) . W e a re going to reca ll the deﬁnition of a similar inv ariant Reg : K 2 ( U ) → H (Γ( U 0 ) , Q ) which is p er haps b es t to call the tame regulator. Let ν : R ∗ → Q b e a v aluation induced by the a bsolute v alue | · | . The nor malization map identiﬁes the irr educible comp onents of U 0 and its nor ma lization which we will use without further notice. A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 23 F or ev ery vertex v ∈ V (Γ( U 0 )) let U v denote op en aﬃne subv ariety of U 0 consisting of irr educible comp onent v with a ll singular p oints remov ed. L et U v denote als o the unique op en aﬃne forma l subscheme of U whos e ﬁber in U 0 is equal to U v . Let U v denote the op en a ﬃnoid of the r igid analytic space U we g e t b y applying Raynaud’s functor to the admissible formal scheme U v . Then U v is a connected rationa l sub do main of P 1 ( C ) via the embedding of U into the latter whic h is isomorphic to U ( S ) for s o me ﬁnite subset S ⊂ f . As we saw in Deﬁnition 4.5 there is a v aluatio n on the ﬁeld M ( U v ), henc e on the ﬁeld M ( U ), whose restriction to R is ν . Let T v denote the corr esp onding gene r alized tame symbol from K 2 ( U ) into the m ultiplicative gro up of the function ﬁeld of U v tensored with Q . F or ev ery k ∈ K 2 ( U ) and every e ∈ E (Γ( U 0 )) let Reg( k )( e ) ∈ Q d enote the v aluation of T o ( e ) ( k ) a t the imag e of e with r esp ect to the normalizatio n ma p. It is not diﬃcult to see that Reg( k ) is a harmonic cochain but this fact a lso follo ws from the following result: Theorem 5 .13. F or every k ∈ K 2 ( U ) we have: Reg( k ) = ν ( { k } ) . Pro of. F or every v ∈ V (Γ( U 0 )) there is a bijection b v from the set B v = { e ∈ E (Γ( U 0 )) | o ( e ) = v } to the set ∂ U v such tha t ν ( { k } b v ( e ) ) = Reg( k )( e ) by Pr op osition 4 .6 . Since for every e ∈ B v we hav e Γ( U 0 ) = b v ( e )( ∂ U ) the claim is now o bvious.  Deﬁnition 5 . 14. Let X b e a geometrica lly co nnec ted reg ular pro jective cur ve deﬁned ov er the ﬁeld R and let R ( X ) denote the ﬁeld of rational functions o f the curve X . F or any x ∈ X ( C ) and any tw o non-zero functions f , g ∈ R ( X ) let { f , g } x denote the tame symbol o f the pa ir ( f , g ) at x . W e deﬁne K 2 ( X ) as the int ersectio n of the k ernels of all the tame symbols {· , ·} x inside K 2 ( R ( X )) where x runs through the se t X ( C ). B y the us ua l abuse of notatio n let X denote a lso the rigid analytic v ariety a sso ciated to the pro jective curve X as w ell. Deﬁnition 5.15. Recall that X is called a Mumford cur ve if ther e is a ﬂat, pro jec- tive, reg ula r and semistable scheme X ov er the sp ectrum o f O whose ge ne r ic ﬁb er ov er R is iso morphic to X and whose sp ecial ﬁb er X 0 ov er f is totally deg e nerate. Let p : U → X b e the univ ersal cov er of X in the catego ry of admissible formal schemes and let Γ denote the group o f deck transforma tions of the cover p . Accord- ing to [6] the formal scheme U is of the t yp e co nsidered in Deﬁnition 5.1 0, at least under some assumptions on the base ﬁeld R . Let p 0 : U 0 → X 0 denote the sp ecial ﬁber of p ov er f and let p : U → X denote the map o f rigid analytic spaces we get by applying Raynaud’s functor to p . The map p 0 induces a ma p of o riented gra phs Γ( U 0 ) → Γ ( X 0 ) which in turn induces a map: p ∗ 0 : H (Γ( X 0 ) , R ∗ ) → H (Γ( U 0 ) , R ∗ ) Γ where the super s cript Γ denotes the subgro up of Γ-inv ariant har monic co chains. The natural action o f Γ on the graph Γ( U 0 ) is prop er and free therefor e p ∗ 0 is in fact an isomorphism. By the inv ariance theorem the regula tor of the pull-back p ∗ ( k ) of a ny element k ∈ K 2 ( X ) with resp ect to p lies in H (Γ( U 0 ) , R ∗ ) Γ . Hence we may deﬁne the rig id analytic regula tor for X a s a map: {·} : K 2 ( X ) → H (Γ( X 0 ) , R ∗ ) given by the rule { k } = ( p ∗ 0 ) − 1 ( { p ∗ ( k ) } ) for every k ∈ K 2 ( X ). 24 AMBRUS P ´ AL Example 5.16 . A c ase of particular interest is when X = E is a T ate elliptic cur ve. In this case the r egulator is uniquely determined by its v alue on any of the edges of the r eduction gr aph of a minimal model of the elliptic curve E ov er Sp ec( C 0 ) so it is its rea lly a homomor phism {·} : K 2 ( E ) → C ∗ , well-deﬁned up to s ign. It is very easy to give an explicit desc r iption of this homomo rphism in general using the T ate uniformiza tion of the elliptic cur ve. Recall that an elliptic cur ve E deﬁned ov er C is a T ate curve if its j inv ariant j ( E ) is not an e le men t o f C 0 . Under this assumption there is a rigid ana lytic T a te uniformization u : C ∗ → E . The pull-back u ∗ ( k ) of every k ∈ K 2 ( E ) as an e le men t o f K 2 ( R ( E )) in K 2 ( M ( U )) for a n y U ⊂ C ∗ connected ra tional sub domain lies in K 2 ( U ) hence the r egulator { u ∗ ( k ) } D ∈ C ∗ is well-deﬁned for ev ery D ∈ ∂ U . The v alue of this regulator { u ∗ ( k ) } D ∈ C ∗ do es not depe nd on the choice of U or D if the disk D contains 0 by the homotopy inv aria nce of the r e gulator. This v alue is the regulato r { k } of the element k ∈ K 2 ( E ). Next we present a purely analytical proo f of W eil’s Recipro city Law for T ate elliptic curves. Theorem 5.17. L et E b e an el liptic curve deﬁne d over C such that its j -invariant j ( E ) / ∈ C 0 and let f , g b e two non-zer o r ational funct ions on E . Then t he pr o duct of al l t ame symb ols of t he p air ( f , g ) is e qual to 1: Y x ∈ E ( C ) { f , g } x = 1 . Pro of. This ar g ument can b e ge neralized to Mumford curves using the co ncept of a fundamen tal do main for a Schottky gro up, but for the s ake of simplicity we present the argument for T ate curves only . As we a lr eady no ted there is a rig id analytic T ate uniformizatio n u : C ∗ → E with T ate p er io d t ∈ C such that | t | > 1. Let f and g a lso denote the pull-back of these functions to C ∗ via u b y abuse of notation. Then the restrictio n of f , g to the annulus A (1 , | t | ) is meromor phic. Let S ⊂ A (1 , | t | ) b e a ﬁnite set such that the functions f and g don’t have a zero or a po le o n A (1 , | t | ) − S . F or each s ∈ S ta ke an op en disk D s ⊂ A (1 , | t | ) containing s such that these disks ar e pair- wise disjoint. The s e t Y = A (1 , | t | ) − S s ∈ S D s is a connected ratio nal sub domain. If these dis ks are s uﬃcie ntly small claim ( iii ) o f Theorem 3.2 r eads as follows: { f , g } D (0 , 1) · { f , g } D ( ∞ , | t | − 1 ) · Y s ∈ S { f , g } s = 1 . But the functions f , g are per io dic with multiplicative p erio d t , so the regulato rs { f , g } D (0 , 1) and { f , g } − 1 D ( ∞ , | t | − 1 ) = { f , g } D (0 , | t | ) are equal, b ecause they only dep end on the r estrictions o f f and g to the se ts ∂ D (0 , 1) and ∂ D (0 , | t | ), resp ectively . Hence the pro duct of the ﬁrst tw o terms in the eq uation ab ov e is one, and the claim follows.  6. The rigid anal ytic regula tor on Drinfeld’s upper half pl a ne Example 6.1. Let R denote again a clo sed subﬁeld of C a nd assume that the v aluatio n on R induced by | · | is discrete. Also assume that the residue ﬁeld of R is a ﬁnite ﬁeld F q and let O deno te the v alua tion ring of R . Let Ω deno te A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 25 the rigid analytic upper half plane, or Drinfeld’s upper half plane ov er R . It is the rigid a nalytic space Ω K int ro duced in Deﬁnition 5.3 in the specia l case when K = P 1 ( R ). In particula r the set o f p oints of Ω is P 1 ( C ) − P 1 ( R ), denoted also b y Ω by abuse of notation. W e can give a very simple description of the r egulator of every k ∈ K 2 (Ω) as follows. By the inv a riance theore m the v alue { k } ( ρ ) = { k | Ω( S ) } D ( ρ ) , where ρ ∈ GL 2 ( R ) and D ( ρ ) ∈ S ∈ P , is indep endent of the choice o f S . W e deﬁne the regulator { k } : GL 2 ( R ) → C ∗ of k as the function given by this rule. The assignment k → { k } is GL 2 ( R )-equiv ar iant by the in v a riance theorem. Deﬁnition 6.2. W e say that an additive s ubmo dule A ⊂ R is a lattice if it is discrete and the quotient A \ R is compact. Le t Γ( A ) denote the following subgroup: Γ( A ) = {  1 a 0 1  ∈ GL 2 ( R ) | a ∈ A } . The s ubg roup Γ( A ) stabilizes the p oint ∞ on the pro jective line via the M¨ obius action. Also note that Γ( A ) leaves the set Ω c = { z ∈ Ω | c < | z | i } inv ar iant for an y p ositive c ∈ | C | where | z | i = inf x ∈ R | z − x | is the imag inary absolute v alue. Let µ be a Haar measure on the additiv e gro up o f the non-archimedean ﬁeld R . This measure induces another measur e on the q uotient g roup A \ R which will be denoted my the sa me s ymbol. W e may no rmalize µ such that µ ( A \ R ) = 1. In this case µ will take only rational v alues. W e may a nd we will ass ume that the absolute v alue | · | o n R is no rmalized suc h that µ ( y O ) = | y | µ ( O ) for every y ∈ R ∗ . If k ∈ K 2 (Ω) is a Γ( A )-inv ariant element then the regula to r { k } : GL 2 ( R ) → C ∗ is also inv ariant with resp ect to the left r egular action o f Γ( A ). Moreov er he regula tor { k } is left inv ariant b y m ultiplication on the r ight by a compact, op en subgroup o f GL 2 ( R ) hence the in tegra nd of the in tegra l { k } ∞ = Z A \ R { k }  1 x 0 1  dµ ( x ) ∈ R ∗ ⊗ µ ( { k } (  1 · 0 1  ) is lo cally constant and the in tegra l itself is well-deﬁned. The mo dulus ab ov e is a subset of Q as we alre ady remarked. Deﬁnition 6.3. Let A 0 denote the intersection A ∩ O and let p de no te the char- acteristic o f the res idue ﬁeld F q . The s e t A 0 is ﬁnite and it is also a vector space ov er F p so its c ardinality is a p ow er o f p . Let e A ( z ) : Ω → C ∗ denote the clas sical Carlitz-exp onential: e A ( z ) = z Y 0 6 = λ ∈ A  1 − z λ  . It is well known (see for ex ample 2 .7 of [5 ], pa ge 44- 4 5) that the function e − 1 A is Γ( A )-inv a riant and it is a biholomor phic map b etw een the quo tient Γ( A ) \ Ω c and a s mall op en disk around 0 punctured at 0 for a suﬃcien tly large c . W e say that a Γ( A )-inv a riant meromorphic function u on Ω is mer o morphic a t ∞ if the comp osition of u and the in verse o f the biholo morphic map e − 1 A is meromo r phic at 0 for some (and hence all) such c num b er. In this ca se we can s p ea k ab out its v alue, order of zero or order of p ole at ∞ . Let M A (Ω) de no te the ﬁeld of Γ( A )- inv ar iant meromorphic functions on Ω mero morphic at ∞ . Let K A 2 (Ω) denote the int ersectio n K 2 (Ω) ∩ K 2 ( M A (Ω)). F or ev ery k ∈ K 2 ( M A (Ω)) we may sp eak a bo ut its tame symbol at ∞ in the sense introduced ab ove. 26 AMBRUS P ´ AL Theorem 6.4. F or e ach element k ∈ K A 2 (Ω) we have µ ( { k } (  1 · 0 1  )) ⊆ Z and t he inte gr al { k } ∞ is e qual t o the tame s ymb ol of k at ∞ mult iplie d by | A 0 | . Pro of. F or every positive c ∈ | C | let D c denote the s e t of those disks D ∈ D such that there is an element S ∈ P such that D ∈ S and Ω( S ) ⊂ Ω c . The se t Ω c has the structure of a r igid analytic s pace such that a function f : Ω c → C is holo morphic if and only if the restriction of f onto Ω ( S ) is ho lomorphic for every S ∈ P whenever Ω( S ) ⊂ Ω c . Let M (Ω c ) denote the ﬁeld of meromor phic functions on Ω c . F or each k ∈ K 2 ( M (Ω)) the v alue { k } ( D ) = { k | Ω( S ) } D , wher e D ∈ D c , D ∈ S ∈ P and Ω( S ) ⊂ Ω c , is indep endent of the choice of S and deﬁnes a function { k } : D c → C ∗ . F or every y ∈ R ∗ and x ∈ R we ha ve D (  y x 0 1  ) = { z ∈ C || z − x | > | y |} , so D (  y x 0 1  ) ∈ D c if | y | > c . In par ticula r the int egra l { k } y ∞ = Z A \ R { k } ( D (  y x 0 1  )) d ( | y | − 1 µ )( x ) ∈ R ∗ ⊗ | y | − 1 µ ( { k } (  y · 0 1  ) is well-deﬁned for every y ∈ R ∗ and k ∈ K 2 ( M (Ω)) when | y | > c and k can b e represented as the linear combination of sy m b ols of Γ( A )-inv a riant meromor phic functions o n Ω c . Under this notation we hav e { k } ∞ = { k } 1 ∞ . W e say that the Γ( A )-in v a riant mero mo rphic function u on Ω c is meromorphic at ∞ if it satisﬁes the same conditio n as we demanded for the e le ments of M A (Ω) in Deﬁnition 6.3. Let M A (Ω c ) deno te the ﬁeld of Γ( A )-inv ariant meromorphic functions on Ω c mero- morphic at ∞ . Let K A 2 (Ω c ) denote the intersection K 2 (Ω c ) ∩ K 2 ( M A (Ω c )). Lemma 6.5 . F or e ach element k ∈ K A 2 (Ω c ) we have | y | − 1 µ ( { k } (  y · 0 1  )) ⊆ Z and { k } y ∞ do es not dep end on the choic e of y ∈ R ∗ wher e | y | > c . Pro of. Fix a unifor mize r π ∈ R . As D (  y x 0 1  ) = D (  y x 0 1  ·  u v 0 1  ) for every u ∈ O ∗ and v ∈ O , we may assume that y = π m for so me in teger m using this identit y when v = 0. On the other hand the identit y ab ov e also implies when v = 1 that the int egra nd of the integral { k } y ∞ is trans la tion-inv ariant with re s pe c t to the g roup y O . Since the measure of the pro jection of this g roup into A \ O with resp ect to the measure | y | − 1 µ is an integer w e get that | y | − 1 µ ( { k } (  y · 0 1  )) ⊆ Z as claimed. The function { k } satisﬁes the identit y : { k } ( D ( g )) = X ǫ ∈ F q { k } ( D ( g  π ǫ 0 1  )) for all g ∈ GL 2 ( R ) b ecause the disks D ( g  π ǫ 0 1  ), ǫ ∈ F q give a pair-wis e disjoint partition of the disk D ( g ). An immediate conseque nc e o f this identit y is the formula: { k } ( D (  π m x 0 1  )) q m µ ( x + π m O ) = X ǫ ∈ F q { k } ( D (  π m +1 x + π m ǫ 0 1  )) q m +1 µ ( x + π m ǫ + π m +1 O ) A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 27 which holds for every x ∈ R . Hence by the translation-inv aria nce of the mea sure q m +1 µ we have: Z A \ R { k } ( D (  π m x 0 1  )) d ( q m µ )( x ) = Z A \ R { k } ( D (  π m +1 x 0 1  )) d ( q m +1 µ )( x ) for every suﬃciently small integer m as claimed.  Let us re tur n to the pro of of theo rem. Cho ose a pr esentation k = P i f i ⊗ g i where f i , g i ∈ M A (Ω). Ther e is a p o sitive c ∈ R such that the r e striction of the functions f i and g i onto the rigid analytic spa ce Ω c are in vertible. F o r ev ery elemen t of K 2 ( M A (Ω c )) we may sp eak ab out its tame sy mbo l at ∞ in the s ense intro duced ab ov e. By Lemma 6.5 it will be suﬃcient to prov e that the tame symbol of k | Ω c at ∞ is e q ual to the integral { k } y ∞ for some y ∈ R ∗ with the prop erty | y | > c . Therefore b y bilinear ity it will b e suﬃcient to prove tha t the tame symbo l o f f ⊗ g at inﬁnit y is e qual to { f ⊗ g } y ∞ for every pair o f functions f , g ∈ O ∗ (Ω c ) b ecause of our assumption on c . In fact it will b e suﬃcient to prov e this cla im in the following three cases: ( i ) the functions f , g a re non-zer o at ∞ , ( ii ) the function f is non-zer o at ∞ and g = e A , ( iii ) b oth f a nd g a r e equal to e A . In the ﬁr st cas e we need to show that { f ⊗ g } y ∞ = 1. W e are going to s how that for every p o s itive ǫ there is a n y ∈ R ∗ with the pr op erty | y | > c s uch that { f ⊗ g } y ∞ ∈ U ǫ . This is s uﬃcient to prov e the claim in the ﬁr st case by Lemma 6.5. Let f ( ∞ ) and g ( ∞ ) ∈ R ∗ denote the v alue of the functions f and g at ∞ , resp ectively . Then the v alues o f the functions f ( z ) /f ( ∞ ) a nd g ( z ) /g ( ∞ ) on the rigid space Ω d are in the set U ǫ for a suﬃciently la r ge d > 0 as the set Ω d maps to a small neighborho o d of 0 with res p ect to e − 1 A . Choo se an element y ∈ R ∗ such that | y | > d . F or every x ∈ R let S ( x ) ∈ P b e a set such that D (  y x 0 1  ) ∈ Ω( S ( x )) and Ω( S ( x )) ⊂ Ω d . By our a ssumptions the holomo rphic functions f /f ( ∞ ), g /g ( ∞ ) are in O ǫ (Ω S ( x ) ) for any x ∈ R hence by Theorem 2.2 we have: { f ⊗ g } (  y x 0 1  ) = { f f ( ∞ ) ⊗ g } (  y x 0 1  ) · { f ( ∞ ) ⊗ g g ( ∞ ) } (  y x 0 1  ) ·{ f ( ∞ ⊗ g ( ∞ ) } (  y x 0 1  ) ∈ U ǫ where we also used that the thir d fac tor o n the right hand side is e qual to 1 . Hence the integral { k } y ∞ is an element of U ǫ , to o , since the co rresp onding modulus is a subset of Z . In the second cas e we need to show that { f ⊗ e A } 1 ∞ = f ( ∞ ) −| A 0 | where f ( ∞ ) denotes a gain the v a lue of the function f at ∞ . It is clea r that { ( f /f ( ∞ )) ⊗ e A } y ∞ = 1 b y rep eating the ar gument use d in the proo f o f the claim in the ﬁrst case, therefore w e ma y assume that f = f ( ∞ ) is constant. By the deﬁnition of the degree homomo rphism w e hav e { f ( ∞ ) ⊗ e A } ( D ) = f ( ∞ ) deg( e A | Ω( S ) )( D ) , where D ∈ D c , D ∈ S ∈ P and Ω( S ) ⊂ Ω c . In fact fo r any u ∈ O ∗ (Ω c ) the expression deg ( u | Ω( S ) )( D ) is indep endent of the choice of S and deﬁnes a function 28 AMBRUS P ´ AL deg( u ) : D c → Z . (It is not diﬃcult se e that this is the v an der P ut loga rithmic diﬀerential when the domain of deﬁnition of u is Ω.) Hence it will suﬃcient to prov e that Z A \ R deg( e A ) dµ ( x ) = −| A 0 | ∈ Z ⊗ Z = Z . By ( ii ) of Theo r em 3.6 we have: deg( e A )( D ( g )) = −|{ λ ∈ A | λ / ∈ D ( g ) }| for every g ∈ GL 2 ( R ) such that ∞ ∈ D ( g ). As ∞ ∈ D (  1 x 0 1  ) = { z ∈ P 1 ( C ) | 1 < | z − x |} for any x ∈ R , we g et: deg( e A )(  1 x 0 1  ) = −|{ λ ∈ A || λ − x | ≤ 1 }| =  −| A 0 | , if x ∈ A + O , 0 , otherwise, so the claim is now clear in the seco nd ca se. In the last case we need to show tha t { e A ⊗ e A } 1 ∞ = ( − 1 ) −| A 0 | . Note that { f ⊗ f } D = ( − 1 ) deg( f )( D ) for every U ⊂ P 1 rational sub domain, D ∈ ∂ U b oundary component and f ∈ O ∗ ( U ) function. T his is o b viously tr ue for rational functions and the gene r al case follows from this one by approximation. Therefor e w e only hav e to show that Z A \ R ( − 1) deg( e A ) dµ ( x ) = ( − 1) −| A 0 | ∈ R ∗ ⊗ Z = R ∗ . It is clear from the ca lculations ab ov e that the int egra nd on the left hand side ab ov e is equal to ( − 1 ) −| A | 0 , if x ∈ A + O , and it is equal to 0, o therwise, he nce the required identit y obviously holds.  References 1. A. A. Beilinson, Higher r e gulators and values of L - functions , [translation], Itogi Nauki i T ek. 24 (1984), 181–238. 2. S. Bosch and W. L ¨ utke b ohmert, F ormal and rigid ge ometry I. Rigid sp ac e s , Math. Ann. 29 5 (1993), 291–317. 3. R. F. Coleman, Dilo garithms, r e gulators and p - adic L -functi ons , Inv en t. Math. 69 (1982) , 171–208. 4. J. F resnel and M. v an der Put, G´ eometrie analytique rigide e t applic ations , Birkh¨ auser, Boston, 1981. 5. E.-U. Gekeler and M. Rev ersat, Jac obians of Drinfeld mo dular curves , J. reine angew. Math. 476 (1996), 27–93. 6. L. Gerritzen and M. v an der Put, Schottky gr oups and Mumfor d c urve s , Lect. Notes Math. 817, Spri nger-V erlag, Berl in-Heidelb erg-New Y ork, 1980. 7. R. M . Hain, Classic al p olylo garithms , Motives (U. Jannsen, S. Kleinman, J.-P . Serre, eds.), v ol. 2, American Math. So c., Providence, Rho de Island, 1994, pp. 3–42. 8. K. Kato, A gener alization of lo c al c lass ﬁeld the ory using K -gro ups II , J. F ac. Sci. Univ. T oky o 27 (1980), 603–683. A RIGID ANAL YTICAL REGULA TOR FOR THE K 2 OF MUMFORD CUR VES 29 9. K. Kato, R esidue homomo rphisms in Milnor K - the ory , Galois groups and their representa- tions, Kinokuyina and North-Holland, T oky o-Amsterdam-New Y ork, 1983, pp. 153–172. 10. J. Milnor, Intr o duction to algebr aic K -the ory , Princeton Universit y Press, Princeton, 1971. 11. M. v an der Put, Les f onctions thˆ eta d’une c ourb e de M umfor d , Group e d’´ etude d’analyse ultram´ etrique 1981/82, Paris, 1982. 12. J.-P . Serre, Gr oup es alg ´ ebriques et c orps de classes , H ermann, Pa ris, 1959. D epar tm en t o f M a th em atic s, 18 0 Q uee n’ s G ate, I m pe r ia l C o l leg e , L o nd o n S W 7 2 A Z , U ni t ed K in g d o m E-mail addr e ss : a.pal@ imperial.ac. uk

A rigid analytical regulator for the K_2 of Mumford curves

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