The Hecke Algebra $\mathbf T_k$ has Large Index

Let $\mathbf T_k(N)^{\mathrm{new}}$ denote the Hecke algebra acting on newforms of weight $k$ and level $N$. We prove that the power of $p$ dividing the index of $\mathbf T_k(N)^{\mathrm{new}}$ 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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Published 1 January 2004