Schneider's p-adic L-function and the conjecture of Birch and Swinnerton-Dyer

2000 Fiscal Year Final Research Report Summary

• Project/Area Number: 11640048
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Japan Women's University
• Principal Investigator: KURIHARA Akira Faculty of Science Japan Women's University, Professor, 理学部, 教授 (50130737)
• Project Period (FY): 1999 – 2000
• Keywords: automorphic form / elliptic curve / L-function / p-adic field / Mumford curve / the conjecture of Birch and Swinnerton-Dyer / Common Lisp

Research Abstract

Let E be an elliptic curve over Q that is not of type CM.We choose a prime p that is a divisor of the conductor of E.Then we can consider an arithmetic group Γ.⊂ PGL_2(Q_p) and an automorphic form ψof weight 2 w.r.t. Γ on the p-adic upper half plane corresponding to E.Assume Γ is torsion-free for simplicity. Then ψ is realized as a p-adic Poincare series corresponding to an element 1≠γ∈Γ. Take another element 1≠δ∈Γ and choose the coordinate of P^1 so that 0 and ∞ are the fixed points of δ on P^1(Q_p). We define L(s, φ, δ)=Σ_<g∈<δ>＼Γ/<γ>>{χ(g・β)^<1-s>-χ(g・α)^<1-s>}. Here α, β are the fixed points of γ, and for the ratio q_δ of eigenvalues of δ we choose a suitable character χ : Q^X_p/q^Z_δ→1+p^<1+[1/(p-1)]>Z_p : (We have to modify this definition when δand γ are proportional in Γ/[Γ, Γ]) Then, we have (i) L (1, ψ, δ) = 0, (ii) d/(ds)L(s, φ, δ)|_<s=1>=Log_χ(<γ|δ>). Here Log_χ is defined by the relation d/(ds)χ(t)^s=Log_χ(t)χ(t)^s and <|> : Γ/[Γ, Γ]×Γ/[Γ, Γ]→Q^X_p is the Manin-Drinfeld pa iring which gives the p-adic periods of the jacobian variety of the Mumford curve uniformized by Γ. This L-function was introduced by P.Schneider and our purpose is to find a recipe to give an arithmetically nice δ and to establish the Birch-Swinnerton
-Dyer conjecture for this L-function including some numerical evidences.
At first we have to find a recipe to give δ. To do that we looked at an easier version of the above conjecture. Let (|) : Γ/[Γ, Γ]×Γ/[Γ, Γ]→Z be the pairing giving the length of intersection of chains on the quotient by Γ of the Bruhat-Tits building. Then, ord_p(<|>)=(|). Now the simplified conjecture is "∃δ, ∀γ, (γ|δ)=0⇔Q-rank of E is positive." In the elliptic modular case, the path on the upper half plane from 0 to √<-1>. ∞ plays the role for δ. However at present, we could not find a nice recipe for our δ. This means that although δ exists as a linar functional of the space of certain p-adic automorphic forms, we don't know whether δ exists as in a geometric sense as above (or as an element of a definite quaternion algebra).