Derived Hecke algebra and cohomology of arithmetic groups

Abstract

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $\Gamma$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$-adic Galois cohomology groups on $H^{\ast }(\Gamma ,{\mathbb{Q}}_p)$, and formulate the central conjecture: the motivic $\mathbb{Q}$-lattice inside these Galois cohomology groups preserves $H^{\ast }(\Gamma ,\mathbb{Q})$.

Revisions after referee report, including corrections/clarifications around graded commutativity of global derived Hecke algebra (which is known in some cases but not in general)

References in corpus (5)

• Derived Galois deformation rings
• A torsion Jacquet--Langlands correspondence
• Comparison of sheaf cohomology and singular cohomology
• Automorphic cohomology, motivic cohomology, and the adjoint $L$-function
• Modularity lifting theorems

Cited by in corpus (8)

• Coherent sheaves on the stack of Langlands parameters
• Automorphic L-invariants for reductive groups
• Eisenstein cocycles in motivic cohomology
• Hecke operators on topological modular forms
• Gelfand's trick for the spherical derived Hecke algebra
• A Satake homomorphism for the $\mathrm{mod}p$ derived Hecke algebra
• Koszul duality for Iwasawa algebras modulo p
• Critical Lambda-adic modular forms and bi-ordinary complexes