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

Harron, Robert; Xiao, Liang

doi:10.5802/aif.2916

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 ¹ ; Xiao, Liang ²

¹ Department of Mathematics, Keller Hall, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA
² Department of Mathematics, Mathematical Sciences Building, University of Connecticut, Storrs, Storrs, CT 06269, USA

Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2449-2464.

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/aif.2916

Classification : 11F33, 14F40
Keywords: Gauss–Manin connections, Nearly overconvergent modular forms, Eigencurves, Families of $p$-adic modular forms
Mot clés : Connexions de Gauss–Manin, Formes modulaires quasi-surconvergents, courbes de Hecke, Familles $p$-adiques de formes modulaires

Affiliations des auteurs :

Harron, Robert ¹ ; Xiao, Liang ²

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     author = {Harron, Robert and Xiao, Liang},
     title = {Gauss{\textendash}Manin connections for $p$-adic families of nearly overconvergent modular forms},
     journal = {Annales de l'Institut Fourier},
     pages = {2449--2464},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
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     year = {2014},
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     zbl = {06387343},
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     url = {http://www.numdam.org/articles/10.5802/aif.2916/}
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Harron, Robert; Xiao, Liang. Gauss–Manin connections for $p$-adic families of nearly overconvergent modular forms. Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2449-2464. doi : 10.5802/aif.2916. http://www.numdam.org/articles/10.5802/aif.2916/

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