Hecke operators play a fundamental role in understanding the arithmetic properties of modular and automorphic forms. Broadly speaking, one (rather ahistorical) way of viewing the conjectural correspondence of Langlands is as making precise the experimentally observed connection between Hecke eigenvalues (or Fourier coefficients) of modular forms and the action of Frobenius elements on certain ‘associated’ Galois representations

In this talk, I will first give an impressionistic overview of the story as it pertains to the case of modular curves. This circle of ideas culminated in Deligne’s proof of the conjecture of Ramanujan on the absolute values of the tau function.

Following this, I’ll explain some of the difficulties in generalizing Deligne’s proof to more general contexts. Among these difficulties, one comes across the need for a better understanding of the mod-p geometry of Hecke correspondences. I’ll give some idea of work in progress (joint with Si Ying Lee) on how one can approach this using a key construction from geometric representation theory: the Vinberg monoid.