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

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

2008-10-27

arxiv.org/abs/0810.4788v1

Mathematics

Number Theory

10 pages

Scientific paper

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.

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