The Hecke Algebra Tk has Large Index
Frank Calegari, Matthew Emerton
Abstract
Let Tk denote the Hecke algebra acting on newforms of weight k and level N. We prove that the power of p dividing the index of Tk 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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