Paper detail

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

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.