Pollack, David; Pollack, Robert A construction of rigid analytic cohomology classes for congruence subgroups of \(\text{SL}_3(\mathbb Z)\). (English) Zbl 1228.11074 Can. J. Math. 61, No. 3, 674-690 (2009). 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. Cited in 2 ReviewsCited in 5 Documents MSC: 11F75 Cohomology of arithmetic groups 11F85 \(p\)-adic theory, local fields 11F60 Hecke-Petersson operators, differential operators (several variables) Keywords:theorem of Ash and Stevens; overconvergent cohomology; rigid analytic eigenclass; cocycle; algorithm PDFBibTeX XMLCite \textit{D. Pollack} and \textit{R. Pollack}, Can. J. Math. 61, No. 3, 674--690 (2009; Zbl 1228.11074) Full Text: DOI