Sur les représentations $p$-adiques géométriques de conducteur 1 et de dimension 2 de $G_{\Q}$

Autour de l'énumération des représentations automorphes cuspidales algébriques de ${\rm GL}_n$ sur $\mathbb{Q}$ de conducteur $>1$

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 Taïbi 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 re…