The rank of quotients of J0(N)

Abstract

Contents 1. Introduction. Given a large prime N, consider the Jacobian variety J 0 (N) of the modular curve X 0 (N ) = H/ 0 (N). There has been considerable study recently of the rank over Q of these varieties and their quotients, much of it dealing with two questions:• Are there large quotients of J 0 (N) with rank zero over the rationals?• How large a rank can a quotient of J 0 (N) have? See Mazur [9] and Merel [10] for arithmetic approaches to the first question; one consequence of the latter is the existence of a quotient with rank zero and dimension at least N 1/8 (out of a maximum possible dimension of D N = N/12 + ᏻ(1), since N is prime). The progress of Kolyvagin [8] and Gross and Zagier [6] toward the Birch-Swinnerton-Dyer conjecture allows one to investigate these questions through the "analytic rank" of J 0 (N ). This is defined (see, for example, [1]) to be the order of the zero at s = 1 of L(J 0 /Q, s) = f L f (s), where the product is taken over L-functions associated with the Hecke eigenbasis of S 2 ( 0 (N)), the weight-2 holomorphic forms on X 0 (N ) (of which there are D N ). In particular, if one can show that many of the L f (s)'s have order zero or one, then the corresponding quotients provide good answers to each of the questions. Along these lines, Duke [4] has used first and second moments of L-functions to show that at least N/(log N) 2 are nonzero at s = 1, which along with Kolyvagin's results may be used to give a corresponding improvement in the answer to the first question.One can also use Riemann's explicit formula for the L f 's (see Mestre [11]) to control the average order at s = 1, although this approach requires (at least at present) an appeal to the Riemann hypothesis to obtain the desired bounds. In recent papers, Brumer [1] and Murty [12] have used these techniques to show that (assuming the

Non-vanishing of high derivatives of automorphic L-functions at the center of the critical strip

“…In recent years, non-vanishing results for central values of families of L functions and their derivatives have received considerable attention ( [Du,Lu,MM,BFH,Iw,KM1,KM2,VdK1] and others) mainly because of their implications in various areas such as the Birch-Swinnerton-Dyer conjecture, spectral deformation theory, and classical analytic number theory. In this paper, we consider a similar question for the central value of higher derivatives of one such family.…”

Non-Vanishing of High Derivatives of Dirichlet L-Functions at the Central Point

“…Proving non-vanishing of automorphic L-functions and their derivatives at the center of the critical strip by analytical means has received considerable attention in recent years (see [1,3,4,6,7,10]). In this paper, we consider the corresponding problem for L-functions attached to primitive Dirichlet characters of high level q.…”

Twists of Automorphic L-Functions at the Central Point

“…as q → ∞. In the case χ is trivial, Duke's result has been subsequently sharpened by [23,15,11,16] to give a positive proportion of non-zero central values. These results are obtained by calculating the mollified moments of the family L(f, s) f ∈S * 2 (q) .…”