The Hecke Algebra Tₖ has Large Index

الأصل · EN

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.