GL 2 ×GSp 2 L-values and Hecke eigenvalue congruences
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 751-775.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1108
Classification: 11F33,  11F46,  14G10
Keywords: Automorphic representations, Hecke-eigenvalues, congruences, L-values
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.
License: CC-BY-ND 4.0
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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, Volume 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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