On the Up operator acting on p-adic overconvergent modular forms when X0(p) has genus 1

Abstract

In this note 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.