Period Integrals and Rankin–Selberg L-functions on GL(n)

Introduction to zeta integrals and L-functions for GLn

All known ways to analytically continue automorphic L-functions involve integral representations using the corresponding automorphic forms. The simplest cases, extending Hecke’s treatment of GL2, need no further analytic devices and very little manipulation beyond Fourier-Whittaker expansions. [1] Poisson summation is a sufficient device for several accessible classes of examples, as in Riemann...

DERIVATIVES AND L-FUNCTIONS FOR GLn

Rational Period Functions and Cycle Integrals

The existence of such a basis is well-known, and our aim here is to illustrate the effectiveness of using weakly holomorphic forms in providing one. Our main goal is to construct modular integrals for certain rational solutions ψ to (1) for any k ∈ Z made out of indefinite binary quadratic forms. A modular integral for ψ is a periodic function F holomorphic on the upper half-plane H and meromor...

CRITICAL VALUES OF RANKIN–SELBERG L-FUNCTIONS FOR GLn ×GLn−1 AND THE SYMMETRIC CUBE L-FUNCTIONS FOR GL2

In a previous article [35] an algebraicity result for the central critical value for L-functions for GLn × GLn−1 over Q was proved assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize [35, Thm. 1.1] for all critical values for L-functions for GLn×GLn−1 over any number field F while using the period relations of [37] and...

Harmonic Weak Maass Forms, Automorphic Green Functions, and Period Integrals

Arakelov geometry, which is a mixture of algebraic geometry at finite primes (of a number field) and real analysis at infinite primes, was invented by Arakelov [Ar] in 70s to ‘compactify’ an arithmetic variety (see also [Fa2]). It has become a very important part of modern number theory after Faltings’ proof of the Mordell conjecture (see for example [Fa1], [So]) and the celebrated Gross-Zagier...