We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn×GLn′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s, π × π̃), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GLn is determined by a finite numbe...

This article is an introduction to the Petersson trace formula and Kuznetsov trace formula, both of which are now important, standard techniques in analytic number theory. To illustrate their applications to modular forms, we will explain their role in a proof of subconvexity bounds for Rankin-Selberg L-functions L(s, f ⊗ g) on the critical line σ = 1/2, where here and throughout, we write s = ...

We give a narrow zero-free region for standard L-functions on GL(n) and Rankin-Selberg L-functions on GL(m) × GL(n) through the use of positive Dirichlet series. Such zero-free regions are equivalent to lower bounds on the edge of the critical strip, and in the case of L(s, π × ˜ π), on the residue at s = 1. Using the latter we show that a cuspidal automorphic representation on GL(n) is determi...

Let K/F be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky’s classification asserts that generic representations are parabolically induced from quasi-squareintegrable representations. We show, following a method developed by Cogdell and PiatetskiShapiro, that expected equality of the Rankin-Selberg type Asai L-function of a generic representation of GL(n, K), and the Asai L-fun...