p-adic L-functions for Galois deformation spaces and Iwasawa Main Conjecture Tadashi Ochiai (Osaka University) January 2006
Main Reference [1] A generalization of the Coleman map for Hida deformation, the American Journal of Mathematics, 2003. [2] Euler system for Galois deformation, Annales de l institut Fourier, 2005. [3] On the two-variable Iwasawa Main Conjecture for Hida deformations, preprint 2004. Contents of the talk General problems on p-adic L-functions Two-variable p-adic L-functions for Hida families 1
Situation p : fixed odd prime number, Q : the cyclotomic Zp-ext. of Q Γ := Gal(Q /Q) χ 1+pZp (Zp) (χ: p-adic cyclo. char) We recall examples of p-adic L- functions. Example 1. Theorem(K-L, I, C). ψ: Dirichlet Character of Conductor D>0 with (D, p) =1 L p (ψ) Zp[ψ][[Γ]] such that χ r (L p (ψ)) = (1 ψ(p)p r )L(ψ, r) for each r 0 divisible by p 1. 2
Example 2. E: an ellip. curve defined over Q. L(E,s): Hasse-Weil L-function for E. Theorem(M-S). If E has good ordinary red. at p, Then, L p (E) Zp[[Γ]] such that φ(l p (E)) = 1 φ(p) 2 α α s(φ) G(φ 1 ) L(E,φ,1) Ω + E for every finite order char. φ on Γ where s(φ) =ord p Cond(φ), α: p-unit root of x 2 a p (E)x+p = 0 with a p (E) =1+p E p (Fp) G(φ 1 ):Gauss sum,ω + E = E(R) ω E, 3
Translation L p (E) is defined on X := {cont. char s Γ η Q p } = a unit ball U(1; 1) Q p (U(a; r) ={x Q p ; x a p <r}) T := T p (E) Zp[[Γ]]( χ) where χ : G Q Γ Zp[[Γ]] Zp[[Γ]]( χ): free Zp[[Γ]]-module of rank one on which G Q = Gal(Q/Q) acts via χ Then the specialization T φ := T Zp [[Γ]] Z p[φ] atφ X is isomorphic to T p (E) φ. L p (E) is associated to T. 4
Consider the following situations: B: a rigid anal. space over Qp (Mostly, we think of a finite cover of an open unit ball in Q s p ) B = SpfO(B) (If B = X, O(X )=Zp[[Γ]]) T : a family of Galois representations over B (Mostly, T = O(B) d ) P : a dense subset in B such that T x = Hét,p (M x ) G Q at each x P for a certain motive M x which is critical in the sense of Deligne. 5
Recall that A (pure) motive M over Q called critical if the composite H + B (M) C H B(M) C H dr (M) C Fil 0 H dr (M) C is an isomorphism, where H + B (M) is +-part for the action of the complex conj. on the Betti realization H B (M). Deligne s conjecture. L(M, 0)/Ω + M Q, (where Ω + M C is the det. of H B + (M) C Fil 0 H dr (M) C). 6
Examples. M = Q(r): Tate Motive L(M, s) = ζ(s + r) is the Riemann s zeta function. M is critical r =2n or 1 2m with n, m Z >0. M = M f (j): j-th Tate twist of the motive for an eigen cuspform f of weight k 2 L(M f (j),s)=l(f,s + j) Hecke L-funct. for f M f (j) is critical 1 j k 1 7
We call (B, T,P) a geometric triple. For a given (B, T,P), consider: Problem. Is there a function L p (T )onb with p-adic continuity which is characterized by the following interpolation property: L p (T )(x) =N x L(M x, 0)/Ω + M x at each x P? (N x is a certain normalization factor at x) 8
Remarks. normalization factors are related to p-adic periods at x, Euler like factor, Gauss sum etc. We have to specify the algebra R O(B) where L p (T ) is contained. Example. B = X, T = T p (E) Zp[[Γ]]( χ) E: supersingular at p. We have L p (E) with the same interpolation property as ordinary cases. L p (E) is never contained in O(X ), but is contained in a larger ring H 1 O(X ). 9
To give a result convincing to formulate the general conjecture, the following Hida deformations are important. Preparation. Γ : the group of Diamond operators on the tower {Y 1 (p t )} t 1 of modular curves Γ χ 1+pZp (Zp) Y := {cont. char s Γ η Q p } = a unit ball U(1; 1) Q p 10
Hida families. a finite cover B q X Y. T is a family of Galois representations on B which is generically of rank two. (Mostly, T = O(B) 2 ) P consists of x B such that q(x) U = χ j(x) χ k(x) satisfying 1 j(x) k(x) 1 for a certain open subgroup U Γ Γ. 11
(B, T,P) is a geometric triple with the following properties: For each x P, f x : an ordinary eigen cuspform of weight k(x) φ x a finite order character of Γ s. t. T x = Tp (f x )(j(x)) φ x ω j(x). (T p (f): rep. of G Q asso. to f, ω: the Teichmuller character) 12
Known constructions of (candidates of) p-adic L-functions for (B, T,P) are classified into three cases below: Use the theory of complex multiplication. (Only for T with CM/by Katz, Yager, etc) Use the theory of modular symbols (Kitagawa, Greenberg-Stevens, etc) Use the Eisenstein family and Shimura s theory (Panchishkin, Fukaya, Ochiai, etc) 13
L Ki p (T ) O(B) is rather desirable so that U an invertible element in (O(B) O Cp ) O(B) such that L Ki p (T ) U O(B) O C p has the interpolation property: (L Ki p (T )(x) U(x))/Ω + p,x = ( 1) j 1 (j 1)! 1 φ xω j (p)p j 1 a p (f x ) p j 1 s(j) L(fx,φ x ω j,j) a p (f x ) at each x P. Ω +,x 14
Remark. Ω + p,x Cp is the p-adic period at x defined to be the determinant of: H B (M fx ) + B HT Fil 0 H dr (M fx ) B HT. This interpolation uniquely characterizes the ideal (L Ki p (T )) O(B) By using both of Ω + p,x and Ω +,x, the interpolation is balanced so that it is independent of the choice of bases. 15
From our detailed study on Selmer groups for T, we have a wellchosen the algebraic p-adic L- function L alg (T ) defined to be p the characteristic ideal of certain Selmer group for T. Thus, we propose Iwasawa Main Conjecture. (L Ki p p (T )) = (Lalg(T )) (refinement of the conj. by Greenberg) 16
Theorem(O-). We have the interpolation map: Ξ:H/f 1 (Q p, T (1)) O(B) with exp x x x Ξ where T (1) = Hom O(B) (T, O(B)(1)) exp x is the dual exponential map H 1 /f (Q p, T (1)) Q p. 17
Z H 1 (Qp, T (1)): Kato s Euler system element such that exp x = L (p) (f x,φ x,j(x)) (2π 1) j 1 L(f x, 1) Ξ(Z) O(B) satisfies (Ξ(Z))(x) = ( 1) j 1 (j 1)! p j 1 a p (f x ) s(j) 1 φ xω j (p)p j 1 a p (f x ) L(f x,φ x ω j,j) (2π 1) j 1 L(f x, 1) 18
Ξ(Z) O(B) gives the interpolation of the L-values, but the complex period is not optimal. Later we arrived the following modification: Theorem (O-). We have a normalized Z Ki H/f 1 (Q p, T (1)) such that Ξ(Z Ki )=L Ki p (T ). This theorem combined with Euler system theory for Galois deformations gives: Theorem (O-). (L Ki p (T )) (Lalg p (T )) 19