no. 6
Gauss–Manin connections for p-adic families of nearly overconvergent modular forms
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, Volume 64 (2014) no. 6, pp. 2449-2464.

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.

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.

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, Volume 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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