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

Open Access article
 Printed: Jun 2009
  • David Pollack
  • Robert Pollack
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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, 11F85 show english descriptions Cohomology of arithmetic groups
$p$-adic theory, local fields [See also 14G20, 22E50]
11F75 - Cohomology of arithmetic groups
11F85 - $p$-adic theory, local fields [See also 14G20, 22E50]

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