An automorphic generalization of the Hermite-Minkowski theorem

TL;DR

This paper proves finiteness results for automorphic representations with bounded weights and conductors over number fields, generalizing the Hermite-Minkowski theorem using positivity properties of quadratic forms and explicit formulas.

Contribution

It introduces a new automorphic finiteness theorem extending Hermite-Minkowski, with explicit bounds depending on root-discriminants and weights, under certain hypotheses.

Findings

Finiteness of cuspidal automorphic representations with bounded weights and conductors.

Introduction of a sequence r(w) controlling finiteness over number fields.

Conditional bounds assuming GRH involving harmonic numbers and Euler's constant.

Abstract

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.