Gauss–Manin connections for $p$-adic families of nearly overconvergent modular forms

[Connexions de Gauss–Manin pour les families $p$-adiques de formes modulaires quasi-surconvergentes]

Harron, Robert ; Xiao, Liang

Annales de l'Institut Fourier, Tome 64 (2014), p. 2449-2464 / Harvested from Numdam

Résumé

Nous obtenons l’interpolation de la connexion de Gauss–Manin en familles $p$-adiques de formes modulaires quasi-surconvergentes. Ceci donne une famille d’opérateurs différentiels à la Maass–Shimura qui envoie l’espace de formes modulaires quasi-surconvergentes de type $r$ dans celui de formes modulaires quasi-surconvergentes de type $r+1$ et de poids $p$-adique augmenté par $2$. Notre méthode est purement géométrique, elle utlise les constructions géométriques des courbes de Hecke dues à Andreatta–Iovita–Stevens et Pilloni, et devrait donc se généraliser aux groupes de rang supérieur.

Abstract

We interpolate the Gauss–Manin connection in $p$-adic families of nearly overconvergent modular forms. This gives a family of Maass–Shimura type differential operators from the space of nearly overconvergent modular forms of type $r$ to the space of nearly overconvergent modular forms of type $r+1$ with $p$-adic weight shifted by $2$. Our construction is purely geometric, using Andreatta–Iovita–Stevens and Pilloni’s geometric construction of eigencurves, and should thus generalize to higher rank groups.

Détail

Publié le : 2014-01-01

MR 3331170 | Zbl 06387343

DOI : https://doi.org/10.5802/aif.2916
Classification:  11F33,  14F40
Mots clés: Connexions de Gauss–Manin, Formes modulaires quasi-surconvergents, courbes de Hecke, Familles $p$-adiques de formes modulaires

Bibliographie

[1] Andreatta, F.; Iovita, A.; Pilloni, V. On overconvergent modular sheaves and modular forms for $GL_{2/F}$ (preprint, to appear in Israel J. Math., doi:10.1007/s11856-014-1045-8)

[2] Andreatta, F.; Iovita, A.; Stevens, G. $p$-adic families of Siegel modular cuspforms (to appear in Ann. of Math.) | Zbl 06399445

[3] Buzzard, K. Eigenvarieties, $L$-functions and Galois representations, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 320 (2007), pp. 59-120 | MR 2367390 | Zbl 1230.11054

[4] Coleman, R.; Gouvêa, F.; Jochnowitz, N. $E_2$, $\Theta $, and overconvergence, Int. Math. Res. Not. (1995) no. 1, pp. 23-41 | Article | MR 1317641 | Zbl 0846.11027

[5] Darmon, H.; Rotger, V. Diagonal cycles and Euler systems I: A $p$-adic Gross–Zagier formula (to appear in Ann. Sci. Éc. Norm. Supér. (4)) | Zbl 1356.11039

[6] Fargues, L. La filtration de Harder–Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math., Tome 645 (2010), pp. 1-39 | Article | MR 2673421 | Zbl 1199.14015

[7] Katz, N. $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III, Springer, Berlin (Lecture Notes in Mathematics) Tome 350 (1973), pp. 69-190 | MR 447119 | Zbl 0271.10033

[8] Pilloni, V. Overconvergent modular forms, Ann. Inst. Fourier, Tome 63 (2013) no. 1, pp. 219-239 | Article | Numdam | MR 3097946 | Zbl 1316.11034

[9] Urban, E. Nearly overconvergent modular forms (to appear in the Proceedings of conference IWASAWA 2012 held at Heidelberg, available at http://www.math.columbia.edu/~urban/EURP.html) | Zbl 1328.11052

[10] Urban, Eric On the rank of Selmer groups for elliptic curves over $\mathbb{Q}$, Automorphic representations and $L$-functions, Tata Inst. Fund. Res., Mumbai (Tata Inst. Fundam. Res. Stud. Math.) Tome 22 (2013), pp. 651-680 | MR 3156865