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

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.Résumé. Nous obtenons l'interpolation de la connexion de Gauss-Manin en familles padiques 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, 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.

Families of Differential Operators for Overconvergent Hilbert Modular Forms

“…Unfortunately, Katz's ∇ is only defined over the ordinary locus, which does not contain the CM points that are supersingular at p. Thus his method only works if the CM points are ordinary at p, which happens if and only if every prime above p in the totally real field K + splits in K K + . The present work extends ∇ to be defined on the overconvergent loci, using methods inspired by those of [HX14] and using the geometry developed in [AIP15] and [AIP16]. This extension allows Damerell's formula to be used in more general situations, whenever the Eisenstein series is defined at the CM points.…”

“…In Section 2, we present the geometric construction of families of overconvergent Hilbert modular forms on the group G * from [AIP16], before generalizing to nearly overconvergent forms. Then in Section 3, we follow [HX14], using the interplay between connections on vector bundles and connections on principal bundles to interpolate the Gauss-Manin connection to families of nearly overconvergent modular forms of any family of p-adic weights. In Section 4, we transport the results of the previous section to the group G using a criterion similar to the one mentioned above.…”

“…In [HX14], the authors ask whether or not their construction can be adapted to avoid a choice of the Hodge filtration. This work answers that question in the affirmative.…”

Nearly overconvergent Siegel modular forms

“…We believe that our method here works also for Shimura varieties for unitary groups. In [23], a construction of the Gauss-Manin connections for nearly overconvergent forms is given in the GL(2) /Q case, where they consider the action of GL(1) (the Levi subgroup of the Siegel parabolic of GL(2)) instead of that of Lie(GL(2)). Note that besides constructing differential operators acting on nearly overconvergent forms of general p-adic analytic weight, there is another problem of taking the differential operator to a p-adic analytic power.…”

A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

“…We point out that there are other possible constructions of the Maaß-Shimura operator on nearly overconvergent forms which are defined on the whole space N r (N, A(U)) and not only on the part of finite slope. In [33], the authors construct an overconvergent Gauß-Manin connection…”