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

TL;DR

This paper constructs a geometric interpolation of the Gauss-Manin connection across p-adic families of nearly overconvergent modular forms, leading to new differential operators with potential for generalization.

Contribution

It introduces a geometric method to interpolate the Gauss-Manin connection in p-adic families, producing a family of differential operators for nearly overconvergent modular forms.

Findings

Constructed a family of Maass-Shimura type differential operators.

Provided a geometric framework using eigencurve constructions.

Potential for extension to higher rank groups.

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.