On the $U_p$ operator acting on $p$-adic overconvergent modular forms when $X_0(p)$ has genus 1

TL;DR

This paper develops methods to compute the action of the $U_p$ operator on overconvergent $p$-adic modular forms for genus 1 modular curves, providing algorithms and explicit examples.

Contribution

It introduces a construction of Banach bases and algorithms for approximating the $U_p$ operator's characteristic series and eigenvectors in the genus 1 case.

Findings

Constructed Banach bases for overconvergent forms

Developed algorithms for $U_p$ characteristic series and eigenvectors

Presented explicit computational examples

Abstract

In this article we will show how to compute $U_p$ acting on spaces of overconvergent $p$-adic modular forms when $X_0(p)$ has genus 1. We first give a construction of Banach bases for spaces of overconvergent $p$-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the $U_p$ operator and eigenvectors of finite slope for $U_p$, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.