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 $\tilde{G}∕\mathbb{Q}$ 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

Mathematical Subject Classification 2010

Primary: 11F75

Milestones

Received: 27 March 2015

Revised: 20 May 2016

Accepted: 30 May 2016

Published: 19 June 2017