We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as , and , where the prime modulus should, for various reasons, appear in the algebraic part of a critical “tensor-product” -value associated to cuspidal automorphic representations of and . Using special techniques for evaluating -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 , et ) 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 du « produit tensoriel » associée à des représentations automorphes cuspidales de et . En utilisant des techniques spéciales pour évaluer les fonctions avec peu de coefficients connus, nous trouvons des approximations suffisantes pour détecter les diviseurs premiers prédits.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1108
Keywords: Automorphic representations, Hecke-eigenvalues, congruences, L-values
@article{JTNB_2019__31_3_751_0, 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}, doi = {10.5802/jtnb.1108}, zbl = {1444.11076}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/} }
TY - JOUR AU - Jonas Bergström AU - Neil Dummigan AU - David Farmer AU - Sally Koutsoliotas TI - $\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences JO - Journal de théorie des nombres de Bordeaux PY - 2019 SP - 751 EP - 775 VL - 31 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/ DO - 10.5802/jtnb.1108 LA - en ID - JTNB_2019__31_3_751_0 ER -
%0 Journal Article %A Jonas Bergström %A Neil Dummigan %A David Farmer %A Sally Koutsoliotas %T $\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-values and Hecke eigenvalue congruences %J Journal de théorie des nombres de Bordeaux %D 2019 %P 751-775 %V 31 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/ %R 10.5802/jtnb.1108 %G en %F JTNB_2019__31_3_751_0
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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