A study of the local components of the Hecke algebra mod $l$

Abstract:

We use information about modular forms $\bmod l$ to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring $\bmod l$. These results suggest that the local components of the Hecke ring $\bmod l$ are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight $k\bmod l$ as $k$ varies within a fixed congruence class $\bmod l - 1$. As an immediate corollary to some of the above results, we show that when $k$ is sufficiently large, an arbitrary prime $l$ must divide the index of the classical Hecke ring ${{\mathbf {T}}_k}$ in the ring of integers of ${{\mathbf {T}}_k} \otimes {\mathbf {Q}}$.