1998 Fiscal Year Final Research Report Summary
Schneider's padic Lfunction and the conjecture of Birch and SwinnertonDye
Project/Area Number 
09640070

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 Japan Woman's University, Faculty of Science Professor, 理学部, 教授 (50130737)

Project Period (FY) 
1997 – 1998

Keywords  automorphic form / elliotic curve / Lfunction / padic field / Munford Curve / the conjecture of Birch and SwinnertonDye / Common Lisp 
Research Abstract 
Let B be a definite quaternion algebra over Q and p a prime unramified in B.Let GAMMA be a subgroup of BX/QX which is a cocompact discrete subgroup of PGL2(Qp). For simplicity, we assume GAMMA is torsionfree. Let * be an automorphic form of weight 2 with respect to GAMMA, which is a GAMMAinvariant global section of the dualizing sheaf of Mumford's padic upper half plane RHO.Also let delta <double plus> 1 be an element of GAMMA, which is meant to define a padic path from a point chi epsilon RHO to delta ・chi epsilon eQ.Then, we can define so called Schneider's padic Lfunction L(s, * , delta). When * is the padic Poincar* series attached to an element gamma <double plus> 1 in GAMMA, we have L(s, *, delta) = SIGMAgepsilon<delta>*GAMMA/(tau) { (g・beta)^<1s>(g・alpha)^<1s>} if delta and gamma are not proportional in GAMMA/[GAMMA, GAMMA], and L(s, *, delta) = SIGMA_1<double plus>epsilon(delta)*GAMMA/(tau){(g・beta)^<1s>(g・alpha)^<1s>} if otherwise. Here we take the coordinate func
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tion z of RHO^1 so that 0 and * are the fixed points of delta, and alpha and beta are the fixed points of gamma. Also ( ) : Q^*_p/q^Z_<delta>* + 1 + p^<1+[1/(p1)]>Zp is a suitable character, where qdelta is the ratio of two eigen values of delta. Then, clearly L(l, *, delta) = 0. By ManinDrinfeld, we have.d/L(s, *, delta)_s = _1= Log( )((gammadelta)), whereLog() is defined by d/(t)^s= Log()(t)(t)^s and () : GAMMA/[GAMMA,GAMMA]*GAMMA/[GAMMA,GAMMA]*Q^*_p is the pairing of ManinDrinfeld giving the padic periods of the Jacobian variety of GAMMA*RHO. Now assume that GAMMA is "of type GAMMAo(NU)" and * is a common eigen function of theHecke operators with rational eigen values. Let E be an elliptic curve over Q : corresponding to *. Then, similarly to MazurTateTeite^^<, >lbaum, it is natural to ask whether there is a recipe for choosing delta (like the path from 0 to ROO<1>・* in the elliptic modular case) so that an analogue of the conjecture ofBirch and Swinnerton Dyer holds for L(s, *, delta) and E.No definitive numerical evidence was found yet, but we are trying to find such an evidence including thecase whenthe Qrank of E is positive. Less
