$\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences

Jonas Bergström; Neil Dummigan; David Farmer; Sally Koutsoliotas

doi:10.5802/jtnb.1108

{GL}_{2} \times {GSp}_{2}

L

-values and Hecke eigenvalue congruences

Jonas Bergström ¹ ; Neil Dummigan ² ; David Farmer ³ ; Sally Koutsoliotas ⁴

¹ Matematiska institutionen Stockholms universitet 106 91 Stockholm, Sweden
² University of Sheffield School of Mathematics and Statistics Hicks Building Hounsfield Road Sheffield, S3 7RH, U.K.
³ American Institute of Mathematics 600 East Brokaw Road San Jose, CA 95112, U.S.A.
⁴ Department of Physics and Astronomy Bucknell University Lewisburg, PA 17837, U.S.A.

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 751-775.

Résumé
Abstract

Nous trouvons des exemples expérimentaux de congruences entre les valeurs propres des opérateurs de Hecke des représentations automorphes de certains groupes (comme ${GSp}_{2} (𝔸)$ , $SO (4, 3) (𝔸)$ et $SO (5, 4) (𝔸)$ ) dans lesquelles le module est un nombre premier qui doit, pour de diverses raisons, apparaître dans la partie algébrique d’une valeur critique de la fonction $L$ du « produit tensoriel » associée à des représentations automorphes cuspidales de ${GL}_{2} (𝔸)$ et ${GSp}_{2} (𝔸)$ . En utilisant des techniques spéciales pour évaluer les fonctions $L$ avec peu de coefficients connus, nous trouvons des approximations suffisantes pour détecter les diviseurs premiers prédits.

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as ${GSp}_{2} (𝔸)$ , $SO (4, 3) (𝔸)$ and $SO (5, 4) (𝔸)$ , where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” $L$ -value associated to cuspidal automorphic representations of ${GL}_{2} (𝔸)$ and ${GSp}_{2} (𝔸)$ . Using special techniques for evaluating $L$ -functions with few known coefficients, we compute sufficiently good approximations to detect the anticipated prime divisors.

Reçu le : 2019-03-05
Révisé le : 2019-09-09
Accepté le : 2019-10-24
Publié le : 2020-05-06

Zbl

DOI : 10.5802/jtnb.1108

Classification : 11F33, 11F46, 14G10
Mots-clés : Automorphic representations, Hecke-eigenvalues, congruences, L-values

Affiliations des auteurs :

Jonas Bergström ¹ ; Neil Dummigan ² ; David Farmer ³ ; Sally Koutsoliotas ⁴

¹ Matematiska institutionen Stockholms universitet 106 91 Stockholm, Sweden
² University of Sheffield School of Mathematics and Statistics Hicks Building Hounsfield Road Sheffield, S3 7RH, U.K.
³ American Institute of Mathematics 600 East Brokaw Road San Jose, CA 95112, U.S.A.
⁴ Department of Physics and Astronomy Bucknell University Lewisburg, PA 17837, U.S.A.

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jonas Bergstr\"om and Neil Dummigan and David Farmer and Sally Koutsoliotas},
     title = {$\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and {Hecke} eigenvalue congruences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {751--775},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
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     zbl = {1444.11076},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/}
}

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Jonas Bergström; Neil Dummigan; David Farmer; Sally Koutsoliotas. $\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 751-775. doi : 10.5802/jtnb.1108. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/

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