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A higher weight generalization of the Hermite-Minkowski theorem

Posted in
Speaker: 
Gaetan Chenevier
Affiliation: 
Université Paris-Sud
Date: 
Mon, 2018-03-12 15:00 - 16:00
Location: 
MPIM Lecture Hall

Let $E$ be a number field, $N$ an ideal of its ring of integers, and $w \geq 0$ an integer. Consider the set of  cuspidal algebraic automorphic representations of $GL_n$ over $E$ whose conductor is $N$, and whose  ''weights'' are in the interval $\{0,\dots,w\}$ (with $n$ varying). If the root-discriminant of $E$ is less than a  certain explicit function $f$ of $w$, then I show that this set is finite. For instance, we have $f(w)>1$ if, and  only if, $w<24$. Under a suitable form of GRH, we may replace $f(w)$ by $8*\Pi*e^{-\psi(1+w)}$, where  $\psi$ is the classical digamma function.

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