The index of $\mathbb{T}^{\text{an}}$ in $\mathbb{T}$

TL;DR

This paper calculates the index of the anemic Hecke algebra within the full Hecke algebra for prime level N, revealing that the index is determined by weight 1 Katz forms at the same level.

Contribution

It provides an explicit computation of the index for prime level N and links it to the existence of weight 1 Katz forms, offering new insights into the structure of Hecke algebras.

Findings

The index is fully described by weight 1 Katz forms.

Explicit formula for the index in terms of Katz forms.

Connection between algebraic structures and modular forms.

Abstract

For a prime level $N$, we compute the index of the anemic Hecke algebra of weight 2 level $\Gamma_0(N)$ forms inside the full Hecke algebra. We prove that the index is fully described by the presence of weight 1 Katz forms of the same level.