A note on control theorems for quaternionic Hida families of modular forms

TL;DR

This paper extends Greenberg and Stevens' control theorem for modular symbols to quaternionic Hida families, covering both definite and indefinite cases, advancing the understanding of p-adic families of modular forms.

Contribution

It generalizes existing control theorems to quaternionic settings, broadening the scope of p-adic modular form theory.

Findings

Extended control theorems to quaternionic Hida families.

Included both definite and indefinite quaternion algebra cases.

Enhanced understanding of modular symbols in quaternionic contexts.

Abstract

We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.