The first non-vanishing quadratic twist of an automorphic L-series

Details The first non-vanishing quadratic twist of an automorphic L-series The first non-vanishing quadratic twist of an automorphic L-series

2010-08-04

arxiv.org/abs/1008.0839v1

Mathematics

Number Theory

30 pages

Scientific paper

Let pi be an automorphic representation on GL(r, A_Q) for r=1, 2, or 3. Let d be a fundamental discriminant and chi_d the corresponding quadratic Dirichlet character. We consider the question of the least d, relative to the data (level, weight or eigenvalue) of pi, such that the central value of the twisted L-series is nonzero, i.e. L(1/2, pi x chi_d) != 0. For example, let N be the level of pi. Using multiple Dirichlet series, we prove the nonvanishing of a central twisted $L$-value with d << N^{1/2} for GL(1), d << N^1 for GL(2), and d << N^2 for GL(3), the last case assuming that a certain character is quadratic (see Theorem 1.13). We work over Q for simplicity but the method generalizes to arbitrary number fields. We conjecture that in all cases there should be such a twist with d << N^epsilon. This would follow from a Lindelof-type bound for a multiple Dirichlet series which does not have an Euler product, but is constructed from a Rankin-Selberg integral applied to automorphic forms which are eigenfunctions of Hecke operators.

Affiliated with

Also associated with

No associations

Rating

The first non-vanishing quadratic twist of an automorphic L-series does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The first non-vanishing quadratic twist of an automorphic L-series, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The first non-vanishing quadratic twist of an automorphic L-series will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-83238