The mod p derived Hecke algebra and its action on arithmetic cohomology
The cohomology of an arithmetic manifold (a higher-dimensional generalization of modular curves) exhibits a peculiar behavior around the middle degree (called the tempered range): its dimension “looks like” that of an exterior algebra.
Recently, Venkatesh and his collaborators have explained this coincidence by constructing new Hecke operators that commute with the classical ones, and increase the cohomological degree. In particular, they consider cohomology with p-torsion and p-adic coefficients, and use Hecke operators at good primes l \neq p.
In this talk, we will describe similar constructions for “l=p”. In particular, assuming standard conjectures (and with a few stronger assumptions for simplicity), we will explain how some “completely p-adic” Hecke operators are related to an action of a certain global Selmer group on tempered cohomology, and we will speculate on whether this action generates tempered cohomology over its smallest degree.