Paul Garrett

Integral moments of automorphic L-functions

This paper exposes the underlying mechanism for obtaining second integral moments of GL2 automorphic L–functions over an arbitrary number field. Here, moments for GL2 are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results. §

Subconvexity bounds for automorphic L-functions

We break the convexity bound in the t -aspect for L -functions attached to cusp forms f for GL 2 ( k ) over arbitrary number fields k . The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L ( s,f ⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

Moments for L-Functions for GLr×GLr-1

We establish a spectral identity for moments of Rankin–Selberg L-functions on GL r ×GL r − 1 over arbitrary number fields, generalizing our previous results for r = 2.

Values of Archimedean Zeta Integrals for Unitary Groups

We prove that certain archimedean integrals arising in global zeta integrals involving holomorphic discrete series on unitary groups are predictable powers of π times rational or algebraic numbers. In some cases we can compute the integral exactly in terms of values of gamma functions, and it is plausible that the value in the most general case is given by the corresponding expression. Non-vanishing of the algebraic factor is readily demonstrated via the explicit expression. Regarding analytical aspects of such integrals, whether archimedean or p-adic, a recent systematic treatment is [Lapid-Rallis 2005] in the Rallis conference volume. In particular, the results of Lapid and Rallis allow us to focus on the arithmetic aspects of the special values of the integrals. This is implicit in (4.4) (iv) in [Harris 2006]. Roughly, the integrals here are those that arise in the so-called doubling method if the Siegel-type Eisenstein series is differentiated transversally before being restricted to the smaller group. In traditional settings, details involving Fourier expansions would be apparent, but such details are inessential. Specifically, many Fourier-expansion details concerning classical Maas-Shimura operators are spurious, referring, in fact, only to the structure of holomorphic discrete series representations. Of course, the translation to and from rationality issues in spaces of automorphic forms should not be taken lightly.

Natural Boundaries and Integral Moments of L-Functions

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a remedy, we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. This class suggests a notion of integral moments of L-functions.