Equivariant torsion and base change

Lipnowski, Michael

doi:10.2140/ant.2015.9.2197

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Equivariant torsion and base change

Michael Lipnowski

Vol. 9 (2015), No. 10, 2197–2240

DOI: 10.2140/ant.2015.9.2197

Abstract

What is the true order of growth of torsion in the cohomology of an arithmetic group? Let $D$ be a quaternion algebra over an imaginary quadratic field $F$ . Let $E ∕ F$ be a cyclic Galois extension with $Γ_{“ E ∕ F} = 〈 σ 〉$ . We prove lower bounds for “the Lefschetz number of $σ$ acting on torsion cohomology” of certain Galois-stable arithmetic subgroups of $D_{E}^{\times}$ . For these same subgroups, we unconditionally prove a would-be-numerical consequence of the existence of a hypothetical base change map for torsion cohomology.

Keywords

torsion, cohomology, Reidemeister torsion, analytic torsion, Ray–Singer torsion, locally symmetric space, trace formula, base change, equivariant, twisted

Mathematical Subject Classification 2010

Primary: 11F75

Secondary: 11F72, 11F70

Milestones

Received: 13 May 2014

Revised: 21 July 2015

Accepted: 6 October 2015

Published: 16 December 2015

Authors

Michael Lipnowski
Mathematics Department Duke University Duke University, Box 90320 Durham, NC 27708-0320 United States

Vol. 9, No. 10, 2015