Rigorous expansions of modular forms at CM points, I: Denominators

TL;DR

This paper presents an algorithm for rigorously computing power series expansions of weight 2 cusp forms at CM points, focusing on bounding denominators without explicit modular curve models, advancing towards equationless Chabauty methods.

Contribution

It introduces a novel algorithm that bounds denominators in expansions at CM points without needing explicit modular curve models, enabling more rigorous computations.

Findings

Algorithm effectively bounds denominators at CM points.

First rigorous method for expansions without explicit models.

Progress towards implementing equationless Chabauty.

Abstract

We describe an algorithm to rigorously compute the power series expansion at a CM point of a weight $2$ cusp form of level coprime to $6$. Our algorithm works by bounding the denominators that appear due to ramification, and without recourse to computing an explicit model of the corresponding modular curve. Our result is the first in a series of papers toward an eventual implementation of equationless Chabauty.