×

A construction of rigid analytic cohomology classes for congruence subgroups of \(\text{SL}_3(\mathbb Z)\). (English) Zbl 1228.11074

Summary: We give a constructive proof, in the special case of \(\text{GL}_3\), of a theorem of Ash and Stevens which compares overconvergent cohomology to classical cohomology. Namely, we show that every ordinary classical Hecke-eigenclass can be lifted uniquely to a rigid analytic eigenclass. Our basic method builds on the ideas of M. Greenberg; we first form an arbitrary lift of the classical eigenclass to a distribution-valued cochain. Then, by appropriately iterating the \(U_p\)-operator, we produce a cocycle whose image in cohomology is the desired eigenclass. The constructive nature of this proof makes it possible to perform computer computations to approximate these interesting overconvergent eigenclasses.

MSC:

11F75 Cohomology of arithmetic groups
11F85 \(p\)-adic theory, local fields
11F60 Hecke-Petersson operators, differential operators (several variables)
PDFBibTeX XMLCite
Full Text: DOI