$\protect \mathrm{GL}_2\times \protect \mathrm{GSp}_2$ $L$-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

@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, Tome 31 (2019) no. 3, pp. 751-775. doi : 10.5802/jtnb.1108. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1108/

Bibliographie
Cité par

[1] Tsuneo Arakawa Vector valued Siegel’s modular forms of degree two and the associated Andrianov $L$ -functions, Manuscr. Math., Volume 44 (1983), pp. 155-185 | DOI | MR | Zbl

[2] Jonas Bergström; Neil Dummigan Eisenstein congruences for split reductive groups, Sel. Math., New Ser., Volume 22 (2016) no. 3, pp. 1073-1115 | DOI | MR | Zbl

[3] Jonas Bergström; Neil Dummigan; Thomas Mégarbané; Tomoyoshi Ibukiyama; Hidenori Katsurada Eisenstein congruences for $SO (4, 3)$ , $SO (4, 4)$ , spinor and triple product $L$ -values, Exp. Math., Volume 27 (2018) no. 2, pp. 230-250 | DOI | MR | Zbl

[4] Jonas Bergström; Carel Faber; Gerard van der Geer Siegel modular forms of degree three and the cohomology of local systems, Sel. Math., New Ser., Volume 20 (2014) no. 1, pp. 83-124 | DOI | MR | Zbl

[5] Spencer Bloch; Kazuya Kato $L$ -functions and Tamagawa numbers of motives, The Grothendieck Festschrift Volume I (Progress in Mathematics), Volume 86, Birkhäuser, 1990, pp. 333-400 | MR | Zbl

[6] Siegfried Böcherer; Neil Dummigan; Rainer Schulze-Pillot Yoshida lifts and Selmer groups, J. Math. Soc. Japan, Volume 64 (2012) no. 4, pp. 1353-1405 | DOI | MR | Zbl

[7] Siegfried Böcherer; Bernhard Heim $L$ -functions on ${GSp}_{2} \times {GL}_{2}$ of mixed weights, Math. Z., Volume 235 (2000) no. 1, pp. 11-51 | MR | Zbl

[8] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[9] Gaëtan Chenevier; Jean Lannes Automorphic forms and even unimodular lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 69, Springer, 2019 | MR | Zbl

[10] Gaëtan Chenevier; David Renard Level one algebraic cusp forms of classical groups (website, http://gaetan.chenevier.perso.math.cnrs.fr/levelone.html) | Zbl

[11] Gaëtan Chenevier; David Renard Level one algebraic cusp forms of classical groups of small rank, Memoirs of the American Mathematical Society, 1121, American Mathematical Society, 2015 | Zbl

[12] Laurent Clozel Motifs et formes automorphes: applications du principe de functorialité, Automorphic forms, Shimura varieties and $L$ -functions, Vol. I (Perspectives in Mathematics), Volume 10, Academic Press Inc., 1990, pp. 77-159 | Zbl

[13] Pierre Deligne Formes modulaires et représentations $l$ -adiques, Séminaire Bourbaki 1968/69 (Lecture Notes in Mathematics), Volume 179, Springer, 1969, pp. 139-172 | DOI | Zbl

[14] Pierre Deligne Valeurs de Fonctions $L$ et Périodes d’Intégrales, Automorphic forms, representations and L-functions (Proceedings of Symposia in Pure Mathematics), Volume 33, American Mathematical Society, 1979, pp. 313-346 | DOI | Zbl

[15] Fred Diamond; Matthias Flach; Li Guo The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 5, pp. 663-727 | DOI | Numdam | MR | Zbl

[16] Tim Dokchitser Computing special values of motivic $L$ -functions, Exp. Math., Volume 13 (2004) no. 2, pp. 137-149 | DOI | MR | Zbl

[17] Neil Dummigan Symmetric square $L$ -functions and Shafarevich-Tate groups, Exp. Math., Volume 10 (2001) no. 3, pp. 383-400 | DOI | MR | Zbl

[18] Neil Dummigan Symmetric square $L$ -functions and Shafarevich-Tate groups, II, Int. J. Number Theory, Volume 5 (2009) no. 7, pp. 1321-1345 | DOI | MR | Zbl

[19] Neil Dummigan; Bernhard Heim; Angelo Rendina Kurokawa–Mizumoto congruences and degree-8 $L$ -functions, Manuscr. Math., Volume 160 (2019) no. 1-2, pp. 217-237 | DOI | Zbl

[20] Neil Dummigan; Tomoyoshi Ibukiyama; Hidenori Katsurada Some Siegel modular standard $L$ -values, and Shafarevich-Tate groups, J. Number Theory, Volume 131 (2011) no. 7, pp. 1296-1330 | DOI | MR | Zbl

[21] Carel Faber; Gerard van der Geer Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre $2$ et des surfaces abéliennes, I, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 5, pp. 381-384 | DOI | Zbl

[22] Carel Faber; Gerard van der Geer Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre $2$ et des surfaces abéliennes, I, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 6, pp. 467-470 | DOI | Zbl

[23] David W. Farmer; Nathan C. Ryan Evaluating $L$ -functions with few known coefficients, LMS J. Comput. Math., Volume 17 (2014), pp. 245-258 | DOI | MR | Zbl

[24] Jean-Marc Fontaine Valeurs spéciales des fonctions $L$ des motifs, Séminaire Bourbaki 1991/92 (Astérisque), Volume 1991, Société Mathématique de France, 1992, pp. 205-249 (Exp. no. 751) | Numdam | MR | Zbl

[25] Masaaki Furusawa On $L$ -functions for $GSp (4) \times GL (2)$ and their special values, J. Reine Angew. Math., Volume 438 (1993), pp. 187-218 | MR | Zbl

[26] Gerard van der Geer Siegel Modular Forms and Their Applications, The 1-2-3 of Modular Forms (Universitext), Springer, 2008, pp. 181-245 | Zbl

[27] Günter Harder A congruence between a Siegel and an elliptic modular form, manuscript, The 1-2-3 of Modular Forms (Universitext), Springer, 2003, pp. 247-262 | Zbl

[28] Bernhard Heim Pullbacks of Eisenstein series, Hecke-Jacobi theory and automorphic $L$ -functions, Automorphic forms, automorphic representations, and arithmetic (Proceedings of Symposia in Pure Mathematics), Volume 66, American Mathematical Society, 1999, pp. 201-238 | DOI | MR | Zbl

[29] Nobushige Kurokawa Congruences between Siegel modular forms of degree $2$ , Proc. Japan Acad., Volume 55 (1979), pp. 417-422 | DOI | MR | Zbl

[30] Thomas Mégarbané Calcul des traces d’opérators de Hecke sur les espaces de formes automorphes (website, http://megarban.perso.math.cnrs.fr/) | Zbl

[31] Thomas Mégarbané Calcul des opérateurs de Hecke sur les classes d’isomorphisme de réseaux pairs de déterminant $2$ en dimension $23$ et $25$ , J. Number Theory, Volume 186 (2018), pp. 370-416 | DOI | Zbl

[32] Thomas Mégarbané Traces des opérators de Hecke sur les espaces de formes automorphes de ${SO}_{7}$ , ${SO}_{8}$ ou ${SO}_{9}$ en niveau $1$ et poids arbitraire, J. Théor. Nombres Bordeaux, Volume 30 (2018) no. 1, pp. 239-306 | DOI | Zbl

[33] Shin-ichiro Mizumoto Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two, Math. Ann., Volume 275 (1986), pp. 149-161 | DOI | MR | Zbl

[34] Jan Nekovář Selmer Complexes, Astérisque, 310, Société Mathématique de France, 2006 | Numdam | Zbl

[35] Jan Nekovář Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Algebra Number Theory, Volume 7 (2013) no. 5, pp. 1101-1120 | DOI | MR | Zbl

[36] Angelo Rendina Congruences of Saito–Kurokawa lifts and divisibility of degree- $8$ $L$ -values, Ph. D. Thesis, University of Sheffield (U.K.) (2019) | Zbl

[37] Kenneth Ribet A modular construction of unramified $p$ -extensions of $ℚ (μ_{p})$ , Invent. Math., Volume 34 (1976), pp. 151-162 | DOI | MR | Zbl

[38] Michael Rubinstein Computational methods and experiments in analytic number theory, Recent perspectives in random matrix theory and number theory (London Mathematical Society Lecture Note Series), Volume 322, Cambridge University Press, 2005, pp. 425-506 | DOI | MR | Zbl

[39] Takakazu Satoh On certain vector valued Siegel modular forms of degree two, Math. Ann., Volume 274 (1986), pp. 335-352 | DOI | MR | Zbl

[40] Jean-Pierre Serre Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Séminaire Delange–Pisot–Poitou 1969/70, Secrétariat Mathématique, 1969 (Exp. no. 19) | Numdam | Zbl

[41] Sug Woo Shin Galois representations arising from some compact Shimura varieties, Ann. Math., Volume 173 (2011) no. 3, pp. 1645-1741 | DOI | MR | Zbl

[42] Christopher Skinner; Eric Urban Sur les déformations $p$ -adiques de certaines représentations automorphes, J. Inst. Math. Jussieu, Volume 5 (2006) no. 4, pp. 629-698 | DOI | Zbl

[43] H. Peter F. Swinnerton-Dyer On $l$ -adic representations and congruences for coefficients of modular forms, Modular functions of one variable. III (Lecture Notes in Mathematics), Volume 350, Springer, 1973, pp. 1-55 | DOI | MR | Zbl

[44] Rainer Weissauer Four dimensional Galois representations (Astérisque), Volume 302, Société Mathématique de France, 2005, pp. 67-150 | Numdam | MR | Zbl

[45] Wolfram The Mathematica computer algebra system (http://www.wolfram.com/mathematica/) | Zbl

[46] Hiroyuki Yoshida Motives and Siegel modular forms, Am. J. Math., Volume 123 (2001) no. 6, pp. 1171-1197 | DOI | MR | Zbl

Cité par Sources :