An automorphic generalization of the Hermite–Minkowski theorem
| Autor: | Gaëtan Chenevier |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
algebraic number theory
General Mathematics Algebraic number theory Mathematics::Number Theory Automorphic form arithmetic geometry automorphic forms 01 natural sciences Ring of integers Combinatorics L-functions Integer 0103 physical sciences FOS: Mathematics Number Theory (math.NT) Ideal (ring theory) 0101 mathematics Algebraic number Mathematics Mathematics - Number Theory [MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] Galois representations 010102 general mathematics 11R 11F 11M 14G Algebraic number field [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] 11F06 11M41 010307 mathematical physics [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] Hermite–Minkowski theorem |
| Zdroj: | Duke Math. J. 169, no. 6 (2020), 1039-1075 Duke Mathematical Journal Duke Mathematical Journal, Duke University Press, 2020, 169 (6), pp.1039--1075 |
| ISSN: | 0012-7094 |
| Popis: | We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, 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 general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs 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. Comment: 30 pages, 1 table |
| Databáze: | OpenAIRE |
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