Computing Galois representations and equations for modular curves $X_H(\ell)$

TL;DR

This paper develops explicit plane models of modular curves to compute associated Galois representations for higher prime levels than previously achieved.

Contribution

It introduces a method to construct explicit equations for $X_H( ext{ell})$ and computes Galois representations for larger primes.

Findings

Explicit equations for $X_H( ext{ell})$ constructed

Galois representations computed for higher primes

Enhanced computational techniques demonstrated

Abstract

We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.