From merel@math.jussieu.fr Wed Jun 10 09:53:02 1998 Return-Path: Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by ferrari.autobahn.org (8.8.7/8.8.7) with ESMTP id JAA31586 for ; Wed, 10 Jun 1998 09:53:01 -0700 Received: from riemann.math.jussieu.fr (root@riemann.math.jussieu.fr [134.157.13.103]) by shiva.jussieu.fr (8.8.8/jtpda-5.3) with ESMTP id SAA24366 for ; Wed, 10 Jun 1998 18:52:56 +0200 (CEST) Received: from [134.157.98.73] (macmerel.math.jussieu.fr [134.157.98.73]) by riemann.math.jussieu.fr (8.8.7/jtpda-5.2) with SMTP id SAA08405 for ; Wed, 10 Jun 1998 18:52:55 +0200 (MET DST) X-Sender: merel@riemann.math.jussieu.fr Message-Id: Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Date: Wed, 10 Jun 1998 18:55:17 +0100 To: was@bmw.autobahn.org From: merel@math.jussieu.fr (Loic Merel) Subject: Re: Status: RO X-Status: A >Loic, > >> Remark: the analogous problem [on L nonvanishing of the L-function] for >> modular symbol is easily answered. > > You'll have to forgive my ignorance about an area in which you are >expert and help me with the following questions. > > Where can I find the analogous answer for modular symbols, i.e., given >an eigenform in characteristic zero expressed in terms of modular symbols >how can I compute whether or not the corresponding L-function vanishes? I >have a fuzzy recollection that there is a completely standard way to do >this by computing some integral to i\infty [...]. Could you point me in >the right direction? (to literature or whatever.) > > If the eigenform f is Q-rational then nonvanishing of the L-function >implies that the sign of the functional equation is -1. (What is the >analogous statement when the eigenform is not Q-rational.) I think this >means that if f = sum a_j [j] is the expression of f on the supersingular >basis then > > I would like to make a table which has two columns. The first list the >eigenforms expressed on the basis of supersingular points and the second >lists whether or not the L-function vanishes. Sorry for having been so elliptic. I am going to try to be more precise. Let p be a prime number. Consider the complex vector space M of functions h: P^1(Z/pZ)-->C satisfying the relations h(x)+h(xS)=0, h(x)+h(xT)+h(xT^2)=0, and h(x)=h(-x) (*) for every x in P^1(Z/pZ), where S and T are the standard elements of PSL_2(Z) of order 2 and 3 respectively. The space M is "equivalent" to the space of modular forms of weight 2 for Gamma_0(p) in the sense that there is a Hecke action on M and the systems of eigenvalues coincide with the systems of eigenvalues arising from modular forms. Therefore every eigenform can be seen as an element of M (well defined up to multiplication by a scalar). The criterion I have alluded to is the following: The L function of an eigenform corresponding to h in M vanishes if and only if h(0)=0. The connection with modular symbols is the following. There is an identification between Gamma_0(p)\SL_2(Z) and P^1(Z/pZ) which sends the class of the matrix abcd to the class of (c,d). There is a map from Gamma_0(p)\SL_2(Z) to the homology of X_0(p) which sends Gamma_0(p)g to the class of the path from g0 to g\infty. Consider a newform f of weight 2 for Gamma_0(p). The integral from g0 to g\infty of f depends only on Gamma_0(N)g which can be seen as an element of P^1(Z/pZ) via the identification above. Therefore we get a map h P^1(Z/pZ)-->C which in fact satisfies the relations (*) above. So to an eigenform we just associated an element of M. The fact that the L-function of f vanishes is equivalent to the fact that the integral from 0 to \infty vanishes. This integral is precisely the particular value of h at the point 0, hence the criterion above. Unfortunately email is not very convenient to explain the full story. Do not hesitate to ask me more questions. Loic