Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal represent...

Abstract We continue with our study of the noncritical exceptional zeros Katz’ $p$-adic $L$-functions attached to a CM field $K$, following two threads. In 1st thread, we redefine (group-ring-valued) ${\mathcal {L}}$-invariant associated each ${\mathbb {Z}}_p$-extension $K_\Gamma $ $K$ in terms height pairings and interpolate them as varies universal (multivariate) group-ring-valued {L}}$-invar...

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...