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A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$

Published:2009-06-01
Printed: Jun 2009
• David Pollack
• Robert Pollack
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Abstract

We give a constructive proof, in the special case of ${\rm GL}_3$, 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 Classifications: 11F75 - Cohomology of arithmetic groups 11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]