Higher regulators of Fermat curves and values of L-functions

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1 Higher regulators of Fermat curves and values of L-functions Mutsuro Somekawa January 10, Introduction Let X be a smooth projective curve over Q with the genus g and L(H 1 (X), s) be the Hasse-Weil L-function. In [B1], Beilinson defines a regulator map reg D : H 2 M(X, Q(m + 2)) H 2 D(X, R(m + 2)) for an integer m 0, where HM 2 (X, Q(r)) is the motivic cohomology (or the absolute chohomology) and HD 2 (X, R(r)) is the Deligne cohomology. Let N 3 be integers and denote by X N the Fermat curve of exponent N, which is the smooth projective curve given by the affine equation : x N + y N = 1. In this paper, we will construct non-zero element of the image of the regulator map reg D for Fermat curves X N. This element is connected to the Beilinson s conjecture which relates the special values of L-function L(H 1 (X), s) in [B1]. We have the canonical isomorphism HD 2 (X, R(m+2)) = [ HB 1 (X(C), Q) Q R(2π 1) m+1] DR from the Betti cohomology. Here, DR represents the invariant subspace under the action of complex conjugation on X(C) and 2π 1. The Beilinson s conjecture predicts that 1. The regulator map induces the Q-lattice structure of HD 2 (X, R(m + 2)). 2. Define c R /Q by g [ regd H 2 M(X, Q(m + 2)) ] = c g [(H 1B(X(C), Q(2π 1) m+1 ) DR] Then, c L (g) (H 1 (X), m) mod Q. This conjecture is formulated for a projective smooth variety and has shown when X is an cyclotomic field and an elliptic curve in [B], [B1]. In the case of the cyclotomic field, the element of the motivic cohomology or its image of the regulator map is called by cyclotomic element. In view point of l-adic story and Hodge story, the cyclotomic element was studies by Deligne and Beilinson in [B2] and [BD]. We will try to construct the analog of cyclotomic element on the Fermat curve. 2. Premieres 2.1 Let F be a number field and S = SpecF. We denote by Sm/S the category of smooth schemes of finite type over S. We will consider the following three cohomology theories on Sm/S with coefficients F = Q, R, Q l. Let X be a scheme X of Sm/S. 1

2 (i)when F = Q, we set H q (X, Q(r)) := H q M (X, Q(r)) = Grr γk 2r q (X) Q where Gr r γ is the subquotient of γ-filtration on the K-group. (ii)when F = R, we set H q (X, R(r)) := H q D (X, R(r)) = Hq (X(C), R(q) D ) DR which is the hyper cohomology of the complex R(r) D := 0 degree 0 R(2π 1) r degree O 1 degree 2 degree r X Ω 1 X Ω r 1 X 0. (iii)when F = Q l for a prime l, we set H q (X, Q l (r)) := H q et(x, Q l (r)). These cohomology theories satisfy the nice cohomological properties which is formulated in [Gi]. We have the regulator maps between these cohomologies reg D :H q (X, Q(r)) H q (X, R(r)), reg et :H q (X, Q(r)) H q (X, Q l (r)). 2.2 We need the notion of homotopical algebra by Quillen in [Q1]. We will give the interpretation of three cohomology theories from a viewpoint of homotopical algebra. Let S be the category of pointed simplicial sets. For a scheme X of Sm/S, we denote by S (X zar ) the category of sheaves of pointed simplicial sets on X zar. In Brown[Br], Brown and Gersten[BG], they have shown that the category S (X zar ) has a structure as a closed model category. We denote by Ho(S (X zar ))(resp. Ho(S )) the homotopical algebra of S (X zar )(resp. S ). Then, we have a functor RΓ (X, ) : Ho(S (X zar )) Ho(S ). We denote by K X = Z Z BGL(O X ) in Ho(S (X zar )), where Z is the completion functor defined by Bousfield-Kan in [BoK]. Since X is regular, K X coincides with the pointed simplicial sheaf G X = QCoh S induced from the presheaf of pointed simplicial sets U QCoh(U) on X zar. Here, Coh(U) is the abelian category of coherent shaef on open set U X and Q is the Quillen s Q-construction in [Q]. Note that K q (X) = π q RΓ (X, K X ), where π q denote q-th homotopy group of pointed simplicial set. By Gillet s methods in [Gi], we have Chern class maps c D r : K X K(2r, Rπ an R(r) D ) c et r : K X K(2r, Rπ et Q l (r)) in the homotopical algebra, where Rα et : D(X et ) D(X zar ) and Rα an : D(X(C) an ) D(X zar ) are the canonical derived functors and K is the Dold- Puppe s construction D(X zar ) Ho(S (X zar )). Note that H 2r q D (X, R(r)) = π q RΓ (X, K(2r, Rπ an R(r) D ))) and H 2r q et (X, Q l ) = π q RΓ (X, K(2r, Rπ et Q l (r))). The regulator maps reg D and reg et are induced from the Chern class map c D r and c et r. 2.3 For integer n 0, we set the finite ordered set [n] := {0 < 1 < 2 < < n}. Let I be the category whose objects are [n] for n 0 and whose morphisms are all maps with preserving the order of elements of objects. A simplicial scheme 2

3 of Sm/S is a contravariant functor from I to Sm/S. We define an augmented simplicial scheme of Sm/S by a pair of scheme X and a simplicial scheme over X. When X is an augmented simplicial scheme of Sm/S, we denote by X 1 the augmented scheme of X and by X 0 the simplicial scheme of X and by X n the scheme of n-simplices of X 0 for n 0. Let assm/s be the category of augmented simplicial schemes of Sm/S. For X of assm/s, we define the cohomology theories by the following way. We set the objects of Ho(S ) K(X 0 ) := coholimrγ (X n, K Xn ), n I RΓ (X 0, R(r)) := coholimrγ (X n, K(2r, Rπ an R(r) D ))), n I RΓ (X 0, Q l (r)) := coholimrγ (X n, K(2r, Rπ et Q l (r))), n I ) K(X ) := coholim (RΓ (X 1, K X 1 ) K(X 0 ), ( ) RΓ (X, R(r)) := coholim RΓ (X 1, K(2r, Rπ an R(r) D ))) RΓ (X 0, R(r)), ( ) RΓ (X, Q l (r)) := coholim RΓ (X 1, K(2r, Rπ et Q l (r))) RΓ (X 0, Q l (r)).. Here, coholim is th homotopy inverse limit functor constructed by Bousfield- Kan in [BoK]. Then, the cohomologies of X are defined by H 2r q M (X, Q(r)) := Gr r γπ q K(X ), We have the following exact sequence. H 2r q D (X, R(r)) := π q RΓ (X, R(r)), H 2r q et (X, Q l ) := π q RΓ (X, Q l (r)). H n (X ) H n (X 1 ) H n (X 0 ) H n+1 (X ) There exists the regulator maps reg D and reg et between these cohomologies. 3. Constructions 3.1 Let S be a base scheme. We set T = G m 2 S the 2-dimensional torus over S with coordinate functions x, y. We define the augmented simplicial scheme T over S by the following. We denote T i(nd) the non-degenarate i-simplices of T. T 1 = T, T 0(nd) = T x T y, T 1(nd) = T x T y = V (x = y = 1), T i(nd) = if i 0, 1 The boundary maps of T are induced from the canonical imbeddings. Let n Z 1. We define T ( n) the augmented simplicial scheme over S by the following inductive way. T ( 1) = T, T ( n) = tot(t ( n + 1) S T ) for n 2 Here, we take tot(t ( n + 1) S T the total simlicial scheme from the bisimplicial scheme T ( n+1) S T. The 1-part of T ( n) is the 2n-dimensional torus T n = G m 2n. 3

4 Proposition Let X be an augmented simplicial scheme over S. Then there exists a following canonical isomorphism of cohomologies. H q 2n (X, F(r 2n)) = H q (tot(x S T ( n) ), F(r)) If x 1, y 1,, x n, y n are the coordinate functions of T ( n) 1 = T n, then the above isomorphism coincide with the cup product of x 1 y 1 x n y n. Remark In [B2], Beilinson has described the same augmented simplicial scheme S( n) which is constructed from T = G m instead of T = G 2 m. As the above notation, we have H q n (X, F(r n)) = H q (tot(x S S( n) ), F(r)). (n+1)-times {}}{ 3.2 Let n Z 1 and T n+1 = T S S T be the 2n + 2-dimensional torus with coordinate x 0, y 0, x 1, y 1,, x n, y n. From subschemes of T n+1, we want to define an augmented bi-simlicial scheme Y (n) over S whose 1-part is Y (n) 1 = T n+1. For 0 i n, we set the S-scheme and we take the immersion n-times Y (n) {i} = {}}{ T S S T ι {i} : Y (n) (n) {i} Y 1, which is defined by i-th ι {i} = id T S S id T S diag S id T S S id T for 0 i n 1, id T S S id T S {(1, 1)} for i = n where the i-th map is the diagonal map: diag : T T S T ; α (α, α). We regard Y (n) (n) {i} as a simplicial subscheme of Y 1 = T n+1 by the immersions ι {i}. For any subset A {0, 1, n}, we set = Y (n) A i A There exists the canonical isomorphism Y (n) A the non-degenarate k-simplices part Y (n) k(nd) Y (n) {i} Y (n) 1. = T n+1 #A over S. For k Z 0, of Y (n) is defined by Y (n) k(nd) = { #A=k+1 T (r) A if 0 k n otherwise. and the boundary maps are induced form canonical injections. Then Y (n) is a bi-simplicial scheme over S. 3.3 We take a divisor D of T defined by equation xy = 1 and we set an open subscheme U = T \ D. Let t : U T be the canonical immersion. Lemma there exists a weak homotopy equivalence of cohomology theories induced from the following canonical morphism of S-simplicial schemes [ ] Y (n 1) = S S( 1) mapping f iber U S Y (n) j n (n) Y. 4

5 Lemma There exists an isomorphism H p 2n (U, F(q 2n)) = H p (U S Y (n), F(q)) From (3.3.1) and (3.3.2), we know that there exists the following long exact sequence H q 1 (Y (l 1) H q (Y (l), F(r)) Hq 2l (U, F(r 2l)), F(r 1)) H q+1 (Y (l), F(r)) by the assumptions for each cohomology theory. couples, we obtain a spectral sequence E l,q Using the theory of exact 1 = H 2(q+l)+α (U, F(2l + β)) H 2(q+l+n)+α (Y (n), F(2n + β)) (3.3.3) Remark The above spectral sequence is an analog of the Beilinson s work in [B2]. When U = P 1 \{0, 1, }, he has constructed the following spectral sequence concerned with the cyclotomic element. E l,q 1 = H q+l (U, F(l + β)) H q+l+n (Ỹ (n), F(n + β)) Here, Ỹ (n) is the same augmented simplicial scheme which is constructed by the same method. Proposition For n 1, there exists the following commutative diagram of exact sequences. 0 HM 1 (S, Q(2n + 1)) H2n+1 M reg (n) (Y 0 H 1 (S, F(n + 1)) H 2n+1 (Y (n) α, F(2r + 1) M α, Q(2n + 1)) H 1 M (U, Q(1)) reg reg H 1 (U, F(1)) The image of α M is generated by the elements 1 xy, x, y O(U) = HM 1 (U, Q(1)). Remark The same statement for H 2n+2 M, Q(2n + 2)) is not evident. But, there exist the canonical map α M : H 2n+2 (n) M (Y, Q(2n + 2)) (U, Q(2)) which induced from the spectral sequence. H 2 M (Y (n) 3.4 We will construct elements which is an analog of the motivic polylogarithm on P 1 \ {0, 1, }. Definition Let N be an integer N 1. We will construct a polylogarithm element Π N by the following way. (i)when N = 2n + 1 is odd, we define an element Π 2n+1 by Π 2n+1 H 2n+1 (Y (n), Q(2n + 1)) such that α M (Π 2n+1 ) = 1 xy Im(α M ) H 1 M (U, Q(1)) = O(U) Z Q from (3.3.5). (ii)when N = 2n + 2 is even, we define an element Π 2n+2 by Π 2n+2 := Π 2n+1 π (x) H 2n+2 (Y (n), Q(2n + 2)). 5

6 Here, when π : Y (n) U is the canonical projection, π (x) is the image of x O(U) Z Q = HM 1 (U, Q(1)) under the map π : HM 1 (U, Q(1)) H 1 (Y (n), Q(1)). By the projection formula, we know that α M (Π 2n+2 ) = {1 xy, x} K 2 (U) Z Q = HM 2 (U, Q(2)). 4. Results 4.1 Let N 3 be integers and denote by X N the Fermat curve of exponent N, which is the smooth projective curve given by the affine equation : x N +y N = 1. We denote by X N := X N U an affine Fermat curve. We take the canonical injection t N : X N U. We obtain the augmented bi-simplicial scheme Y (n) X N,t N and the canonical morphism t N : Y (n) X N,t N Y (n). The following morphism of the spectral sequences is induced form t N. E l,q 1 = H 2(q+l)+α (U, F(2l + β)) H 2(q+l+n)+α (Y (n), F(2n + β)) E l,q 1 = H 2(q+l)+α (X N, F(2l + β)) H 2(q+l+n)+α (Y (n) X N,t N, F(2n + β)) (4.1.1) We get the following proposition on the affine Fermat curve. Proposition The above bottom spectral sequence on the Fermat curve degenerates at E 1. Proof: The boundary map E l,q 1 E l+1,q 1 is the cup product of the element {x, y} K 2 (X N ) Z Q. In Milnor K-group K 2 (X N ), we know that N 2 {x, y} = {x N, y N } = {x N, 1 x N } = 0. So, the proof is complete. From the above proposition, we get the filtration F q of H 2n+2 (Y (n) X N,t N, F(2n+2)) such that Gr q F = H2q (X N, F(2q). So, this implies that there exists the following element. α n = Π 2n+2 Π 2n x n y n H 2 M(X N, Q(2n + 2)) If reg D (α n ) is not zero, then reg D (α n ) sholud present the value of L-function L(H 1 ( X N ), s) at s = 2n. We will compute the value of reg D (α n ). When n = 0, we have α 0 = {1 xy, x}. In [R], Ross compute the value of reg D {1 xy, x}. 4.2 Let ζ = exp(2ı 1), and let A i,j denote the automorphism of X N (C) given by (x, y) (ζ i x, ζ j y). Let t 1/N denote the principal branch of the N-th root function, and let γ : [0, 1] X N (C) denote the path from (1, 0) to (0, 1) given by γ(t) = (t 1/N, (1 t) 1/N ). For integers m and n, let γ m,n denote the following closed path on X N (C). γ m,n := γ A m,0 γ + A m,n A 0,n γ From the computation of the double complex of the differential modules on X N (C), we obtain the following formula. For α > 0, we define by reg(α 1 ) = log(1 xy)(log x) 2 dx x H2 D(U, R(4)) B m (α) := 1 0 dt t3 m dt t t α 1 (1 t) α dt 1. t m 0 t 2 0 t 1 6

7 When m = 1, This function coincides with the beta function B 1 (α) = 1 2B(α, α). Theorem The element α 1 HM 2 (X N U, Q(4)) has non-zero image under the regulator map reg D : HM 2 (X N, Q(4)) HD 2 (X N, R(4)). Proof: We describe the pairing of reg(α 1 ) and some close path γ = γ 1,1 +γ 1, 1 by using the above functions. We computes that reg(α), γ = 3 8π 3 k=1 For 0 M N 1, we set β M = following. reg(α), γ = [ N 1 2 ] M=1 sin 2πk N sin 2πk N k M ( 1 cos 2πk ) ( ) k B 3 N N mod N ( ) 1 k k B 3. Then, we get the N ( 1 cos 2πk ) (β M β N M ) N Since β 1 > > β N 1, the above is not zero. So, the proof is complete. References [B1] Beilinson,A., Higher regulators and values of L-functions, J.Soviet Math. 30, 1985, p [B2] Beilinson,A., Polylogarithm and cyclotomic elements (preprint), 1989 [BD] Beilinson,A.Deligne,P., Interprétation motivique de la conjecture de Zagier, in Symp. in Pure Math., vol.55, part 2, 1994 [BL] Beilinson,A.,Levin A.M., Elliptic polylogarithms, Symposium in pure mathematics, 1994, vol.55, part 2,p [BK] Bloch,S.,Kato,K., Tamagawa numbers of motives and L-functions, The Grothendieck Festschrift volume I Progress in Mathematics vol.86 Birkhäuser, 1990 p [Bl] Bloch,S., Lecture on Algebraic Cycles, Deke Mathematical Series, Duke University Press, 1980 [BoK] Bousfield,A.K.,Kan,D.M., Homotopy limits, Completions and Localizations, Lecture Notes in Math. vol.304, Springer, 1972 [D] Deligne, Le Groupe Fondamental de la Droite Projective Moins Trois Points, In:Galois Groups over Q. M.S.R.I.Publ. Springer vol 16, 1989 p [Gi] Gillet,A., Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. vol. 40, 1981, p [GL] Goncharov,A.B.,Levin,A.M., Zagier s conjecture on L(E, 2), Inv. math., vol. 132, 1998, p

8 [K] Kimura,K., K 2 of Fermat Quotient and the Values of its L-functoin, K- Theory vol. 10, 1996, p [M] Milnor,J., Introdunction to algebraic K-Theory, Annals of Matematics Studies, 72, Princeton Unipersty Press, 1971 [Q1] Quillen,D., Homotopical algebra, Springer Lecture Notes in Math. vol.43, 1967 [Q2] Quillen,D., Higher K-theory I, Higher K-Theory, Springer Lecture Notes in Math., vol 341, 1973, p [So] Somekawa,M., Log-syntomic regulators and p-adic ploylogarithms, K- Theory 17, 1999, p [S1] Soulé,C., Elements cyclotomiques en K-théorie, Astérisque , 1985, p [T] [R] Terasoma,T., Mixed Tate motives and multiple zeta values, Inv. math. Vol.149, No.2,p , 2002 Ross,R., K 2 of Fermat cureves and values of L-functions, C.R.Acad.Sci. Paris, t.312, Série I, p.1-5,