| Budget Amount *help |
¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥800,000 (Direct Cost: ¥800,000)
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| 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 torsion-free. Let * be an automorphic form of weight 2 with respect to GAMMA, which is a GAMMA-invariant global section of the dualizing sheaf of Mumford's p-adic upper half plane RHO.Also let delta <double plus> 1 be an element of GAMMA, which is meant to define a p-adic path from a point chi epsilon RHO to delta ・chi epsilon eQ.Then, we can define so called Schneider's p-adic L-function L(s, * , delta). When * is the p-adic Poincar* series attached to an element gamma <double plus> 1 in GAMMA, we have L(s, *, delta) = SIGMAgepsilon<delta>*GAMMA/(tau) { (g・beta)^<1-s>-(g・alpha)^<1-s>} 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)^<1-s>-(g・alpha)^<1-s>} 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/(p-1)]>Zp is a suitable character, where qdelta is the ratio of two eigen values of delta. Then, clearly L(l, *, delta) = 0. By Manin-Drinfeld, we have.d/L(s, *, delta)|_s = _1= Log( )((gamma|delta)), whereLog() is defined by d/(t)^s= Log()(t)(t)^s and (|) : GAMMA/[GAMMA,GAMMA]*GAMMA/[GAMMA,GAMMA]*Q^*_p is the pairing of Manin-Drinfeld giving the p-adic 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 Mazur-Tate-Teite^^<, >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 Q-rank of E is positive. Less
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