#### Vol. 289, No. 2, 2017

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Local constancy of dimension of slope subspaces of automorphic forms

### Joachim Mahnkopf

Vol. 289 (2017), No. 2, 317–380
DOI: 10.2140/pjm.2017.289.317
##### Abstract

We prove an analogue of a Gouvêa–Mazur conjecture on local constancy of dimension of slope subspaces of modular forms on the upper half plane for automorphic forms on reductive algebraic groups $\stackrel{̃}{G}∕ℚ$ having discrete series. The proof uses a comparison of Bewersdorff’s elementary trace formula for pairs of congruent weights and does not make use of methods from $p$-adic Banach space theory, overconvergent forms or rigid analytic geometry.

We also compare two Goresky–MacPherson trace formulas computing Lefschetz numbers on weighted cohomology for pairs of congruent weights; this has an application to a more explicit version of the Gouvêa–Mazur conjecture for symplectic groups of rank $2$.

##### Keywords
Gouvêa–Mazur conjecture, cohomology of arithmetic groups
Primary: 11F75
##### Milestones
Received: 27 March 2015
Revised: 20 May 2016
Accepted: 30 May 2016
Published: 19 June 2017
##### Authors
 Joachim Mahnkopf Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 1190 Wien Austria