The Hecke Algebra T k has Large Index

  title={The Hecke Algebra T k has Large Index},
  author={Frank Calegari and Matthew Emerton},
Let Tk(N) 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(N) 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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