The Hecke Algebra T_k has Large Index
Frank Calegari, Matthew Emerton

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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Combinatorial Mathematics · Algebraic Geometry and Number Theory
