2000 Fiscal Year Final Research Report Summary
Schneider's padic Lfunction and the conjecture of Birch and SwinnertonDyer
Project/Area Number 
11640048

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear 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 / Lfunction / padic field / Mumford curve / the conjecture of Birch and SwinnertonDyer / 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 padic upper half plane corresponding to E.Assume Γ is torsionfree for simplicity. Then ψ is realized as a padic 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)^<1s>χ(g・α)^<1s>}. 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/(p1)]>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(<γδ>). 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 ManinDrinfeld pa
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iring which gives the padic periods of the jacobian variety of the Mumford curve uniformized by Γ. This Lfunction was introduced by P.Schneider and our purpose is to find a recipe to give an arithmetically nice δ and to establish the BirchSwinnerton Dyer conjecture for this Lfunction 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 BruhatTits building. Then, ord_p(<>)=(). Now the simplified conjecture is "∃δ, ∀γ, (γδ)=0⇔Qrank 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 padic 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). Less
