Distinguishing eigenforms modulo a prime ideal

TL;DR

This paper establishes a new bound on the number of initial Fourier coefficients needed to determine if two modular eigenforms are congruent modulo a prime ideal, generalizing previous results and improving practical bounds.

Contribution

It introduces a novel bound for eigenforms modulo a prime ideal, extending Sturm's and related results, and generalizes a bound on the least prime in an arithmetic progression.

Findings

New bound for eigenforms modulo prime ideal

Generalization of Bach and Sorenson's prime bound

Improved criteria for modular form congruences

Abstract

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given weight and level. This was adapted by Ram Murty, Kohnen and Ghitza to the case of two eigenforms of the same level but having potentially different weights. We consider their expansions modulo a prime ideal, presenting a new bound. In the process of analysing this bound, we generalise a result of Bach and Sorenson, who provide a practical upper bound for the least prime in an arithmetic progression.