An automorphic generalization of the Hermite–Minkowski theorem

Abstract

We show that, for any integer $N\ge 1$, there are only finitely many cuspidal algebraic automorphic representations of ${\mathrm{GL}}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,\dots ,23\}$. More generally, we define an explicit sequence $\left(\mathrm{r}\right(w){)}_{w\ge 0}$ such that, for any number field $E$ whose root discriminant is less than $\mathrm{r}\left(w\right)$ and any ideal $\mathcal{N}$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of ${\mathrm{GL}}_n$ over $E$, with $n$ varying, whose conductor is $\mathcal{N}$ and whose weights are in the interval $\{0,\dots ,w\}$. We also show that, assuming a version of the generalized Riemann hypothesis, we may replace $\mathrm{r}\left(w\right)$ with $8\pi e^{-\psi (1+w)}$ in this statement. The proofs here 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 $\mathrm{L}$-functions. Both the effectiveness and the optimality of the methods are discussed.