Overconvergent modular forms

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

“…Our construction (which was worked out independently) is purely geometric and hence can be easily generalized to similar situations whenever there is a geometric construction like the ones given by Andreatta-Iovita-Stevens [1] and Pilloni [8].…”

“…the construction of Andreatta-Iovita-Stevens [1] and Pilloni [8]. Then, we obtain our family version of ∂ by applying the formal dictionary between principal bundles and vector bundles associated to the representation of the monodromy group.…”

“…This approach essentially follows from the work of Andreatta, Iovita, and Stevens [1] and, independently, Pilloni [8]. The advantage of this construction is that nearly overconvergent modular forms are on the nose sections of certain vector bundles.…”

“…(4) Technically speaking, just knowing the description of Hodge-Tate map over closed points is not enough to prove the existence of Fn. Pilloni [8] needed a version of the Hodge-Tate map over a formal scheme, just as in [1]. However, this subtlety does not matter to our discussion, so we ignore it.…”

“…Since it is sufficient for their purposes, these authors rely on q-expansions in their definition of nearly overconvergent modular forms. Our approach, via the work of Andreatta-Iovita-Stevens [1] and Pilloni [8], is geometric: we give a geometric definition of nearly overconvergent modular forms of arbitrary p-adic weight and construct p-adic families of Gauss-Manin connections varying over the weight space. This provides a robust theory that is amenable to generalization.…”