The index of a certain quotient of the Hecke algebra in its normalization

TL;DR

This paper investigates the relationship between the index of a certain order in the number field generated by a modular form's Fourier coefficients and primes related to congruences of the form modulo prime ideals, connecting algebraic and modular properties.

Contribution

It establishes a link between the primes dividing the index of the order in the ring of integers and primes where the modular form is congruent to a conjugate modulo those primes.

Findings

Primes dividing the index correspond to congruences of the modular form with conjugates.

The index of the quotient of the Hecke algebra relates to the algebraic structure of Fourier coefficients.

The work connects algebraic properties of orders with modular form congruences.

Abstract

Let $\Gamma$ be a congruence subgroup of $SL_2(Z)$, and let $f$ be a normalized eigenform of weight $k$ on $\Gamma$. Let $K$ denote the number field generated over $Q$ by the Fourier coefficients of $f$. Let $R$ denote the the order in $K$ generated by the Fourier coefficients of $f$, which is contained in the ring of integers $O$ of $K$. We relate the primes that divide the index of $R$ in $O$ to primes $p$ such that $f$ is congruent to a conjugate of $f$ modulo a prime ideal of residue characteristic $p$. The index mentioned above is the same as the index of the quotient of the Hecke algebra by the annihilator ideal of $f$ in its normalization.