The Hecke Algebra T_k has Large Index

TL;DR

This paper proves that the p-adic valuation of the index of the Hecke algebra T_k in its normalization grows at least linearly with the weight k, providing new insights into the algebra's structure.

Contribution

It establishes a linear lower bound on the p-adic valuation of the index of T_k, answering a question posed by Serre and applying methods to conjectures of Buzzard and Mazur.

Findings

p-adic valuation of index grows linearly with k

Provides heuristic evidence for Buzzard and Mazur's conjectures

Answers Serre's question on the growth of the index

Abstract

Let T_k denote the Hecke algebra acting on newforms of weight k and level N. We prove that the power of p dividing the index of T_k inside its normalisation grows at least linearly with k (for fixed N), answering a question of Serre. We also apply our method to give heuristic evidence towards recent conjectures of Buzzard and Mazur.