Computing overconvergent forms for small primes

TL;DR

This paper constructs explicit bases for overconvergent p-adic modular forms at small primes, analyzes their stability, and applies these results to compute slopes of eigencurves and construct special points on elliptic curves.

Contribution

It extends Lauder's algorithms to small primes p=2,3, providing explicit bases and stability analysis for overconvergent forms at these primes.

Findings

Computed slope sequences of 2-adic eigencurves.

Constructed Chow-Heegner points on elliptic curves.

Extended algorithms for overconvergent forms to small primes.

Abstract

In this note, we construct explicit bases for spaces of overconvergent $p$-adic modular forms when $p=2,3$ and study their stability under the Atkin operator. The resulting extension of the algorithms of Lauder is illustrated with computations of slope sequences of some $2$-adic eigencurves and the construction of Chow-Heegner points on elliptic curves via special values of Rankin triple product L-functions.