A note on some $p$-adic analytic Hecke actions

TL;DR

This paper demonstrates that Hecke operators act locally analytically on overconvergent modular forms in the $p$-adic setting, enabling an extension of the Hecke algebra action to rigid functions and confirming a conjecture of Gouva.

Contribution

It establishes the local analyticity of Hecke actions away from $p$ on overconvergent forms and extends the Hecke algebra action to rigid functions, confirming Gouva's conjecture.

Findings

Hecke operators act locally analytically on overconvergent modular forms.

The Hecke algebra action extends to rigid functions on its generic fiber.

Determines Hodge-Tate-Sen weights of associated Galois representations.

Abstract

We show that the action of Hecke operators away from $p$ on the space of ($p$-adic) overconvergent modular forms is ($p$-adically) locally analytic in a certain sense. As a corollary, the action of the Hecke algebra can be extended naturally to an action of rigid functions on its generic fiber. This directly determines the Hodge-Tate-Sen weights of Galois representation associated to an overconvergent eigenform and confirms a conjecture of Gouv\^{e}a.