Autour de l'\'enum\'eration des repr\'esentations automorphes cuspidales alg\'ebriques de ${\rm GL}_n$ sur $\mathbb{Q}$ de conducteur $>1$

TL;DR

This paper classifies cuspidal algebraic automorphic representations of ${ m GL}_n$ over $Q$ with small prime conductor and motivic weight up to 17, using explicit formulas and relating to classical objects.

Contribution

It provides an explicit classification of such representations for conductor 2 and motivic weight up to 17, extending previous work to conductor greater than 1.

Findings

List of all such representations with motivic weight ≤ 17 and conductor 2.

Development of an analytical method based on Riemann-Weil explicit formulas.

Proof of a special case of Gross' conjecture for paramodular invariants.

Abstract

We prove classification results for the cuspidal automorphic algebraic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) of small prime conductor and small motivic weight, in the spirit of the works of Chenevier, Lannes and Ta\"ibi in conductor $1$. The main result is an explicit list of all such representations with motivic weight up to $17$ and conductor $2$. For this, we develop the analytical method based on the Riemann-Weil explicit formulas, and use Arthur's work to relate those representations to classical objects. A key ingredient is a special case of Gross' conjecture regarding paramodular invariants of representations of a split ${\rm SO}_{2n+1}(\mathbb{Q}_p)$, which we prove as well.