no. 6
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, Robert1 ; Xiao, Liang2
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.

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.

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

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 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2916/

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