On the ramification of modular parametrizations at the cusps

TL;DR

This paper studies how modular parametrizations of elliptic curves ramify at cusps, proving conditions for unramified cases and exploring numerical patterns of ramification indices.

Contribution

It establishes a criterion linking minimal level twists to unramified parametrizations at cusps, using Bushnell's formula and numerical evidence.

Findings

Unramified at cusps when the associated modular form has minimal level among twists.

Numerical data suggests ramification indices at cusps divide 24.

Provides a new perspective on the ramification behavior of modular parametrizations.

Abstract

We investigate the ramification of modular parametrizations of elliptic curves over Q at the cusps. We prove that if the modular form associated to the elliptic curve has minimal level among its twists by Dirichlet characters, then the modular parametrization is unramified at the cusps. The proof uses Bushnell's formula for the Godement-Jacquet local constant of a cuspidal automorphic representation of GL(2). We also report on numerical computations indicating that in general, the ramification index at a cusp seems to be a divisor of 24.