arXiv subjects

Paul Garrett

Publications and source records attributed to Paul Garrett.

3 recordsLinked to original sources

arXiv2021

Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $\Gamma \backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle.

math.NT↗

arXiv2020

Designed Pseudo-Laplacians

We elaborate and make rigorous various speculations about the implications of spectral properties of self-adjoint operators on spaces of automorphic forms for location of zeros of $L$-functions. Some of these ideas arose in work of Colin de Verdière, Lax-Phillips, and Hejhal, from the late 1970s and early 1980s, not to mention semi-apocryphal attributions to Pólya and Hilbert. For example, given a complex quadratic extension $k$ of $\mathbb Q$, we give a natural self-adjoint extension of a restriction of the invariant Laplacian on the modular curve whose discrete spectrum, if any, consists of values $s(s-1)$ for zeros $s$ of $ζ_k(s)$. Unfortunately, there seems to be no reason for this discrete spectrum to be large. In fact, Montgomery's pair correlation, and the behavior of $ζ(1+it)$, imply that at most $94\%$ of zeros of $ζ(s)$ can appear in this discrete spectrum. Less naively, some preliminary positive results about the dynamics of zeros do follow from these considerations.

math.NT↗

arXiv2006

Integral Moments of Automorphic L-functions

This paper exposes the underlying mechanism for obtaining second integral moments of $GL_2$ automorphic $L$--functions over an arbitrary number field. Here, moments for $GL_2$ 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 $\Bbb{Q}$, we recover the classical results.

math.NT↗