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

TL;DR

We address how the index $[\mathcal{O}:R]$ attached to the Fourier-coefficient field of a cusp form $f$ reflects congruences between $f$ and its Galois conjugates. The approach reformulates the question via the Hecke algebra $\mathbf{T}$ and its annihilator $I_f$, and analyzes ramification and inertia through the Galois closure ${\widetilde{K}}$ and the discriminant of $\mathbf{T}$. A key technical device is a semisimple $\mathbb{F}_p$-algebra analysis that links primes dividing $[{\widetilde{\mathcal{O}}}:R]$ to congruences with nontrivial automorphisms of the coefficient field. In the Galois setting, the authors obtain an if-and-only-if relation: a prime $p$ ramifying in $\mathcal{O}$ or dividing $[\mathcal{O}:R]$ occurs exactly when $f$ is congruent modulo a prime to a nontrivial Galois conjugate of $f$, with implications for the discriminant behavior of the Hecke algebra.

Abstract

Let $Γ$ be a congruence subgroup of $SL_2(Z)$, and let $f$ be a normalized eigenform of weight $k$ on $Γ$. 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.

Theorems & Definitions (16)

• Proposition 1.1
• proof
• Theorem 1.2
• Corollary 1.3
• proof
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof