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15 April 2020 An automorphic generalization of the Hermite–Minkowski theorem
Gaëtan Chenevier
Duke Math. J. 169(6): 1039-1075 (15 April 2020). DOI: 10.1215/00127094-2019-0049

Abstract

We show that, for any integer N1, there are only finitely many cuspidal algebraic automorphic representations of GLn over Q, with n varying, whose conductor is N and whose weights are in the interval {0,,23}. More generally, we define an explicit sequence (r(w))w0 such that, for 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 GLn over E, with n varying, whose conductor is N and whose weights are in the interval {0,,w}. We also show that, assuming a version of the generalized Riemann hypothesis, we may replace r(w) with 8πeψ(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 L-functions. Both the effectiveness and the optimality of the methods are discussed.

Citation

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Gaëtan Chenevier. "An automorphic generalization of the Hermite–Minkowski theorem." Duke Math. J. 169 (6) 1039 - 1075, 15 April 2020. https://doi.org/10.1215/00127094-2019-0049

Information

Received: 22 February 2018; Revised: 5 June 2019; Published: 15 April 2020
First available in Project Euclid: 14 March 2020

zbMATH: 07198471
MathSciNet: MR4085077
Digital Object Identifier: 10.1215/00127094-2019-0049

Subjects:
Primary: 11F06
Secondary: 11M41

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 6 • 15 April 2020
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