#### Vol. 9, No. 10, 2015

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Equivariant torsion and base change

### Michael Lipnowski

Vol. 9 (2015), No. 10, 2197–2240
##### 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 ${\Gamma }_{“\phantom{\rule{0.3em}{0ex}}E∕F}=〈\sigma 〉$. We prove lower bounds for “the Lefschetz number of $\sigma$ acting on torsion cohomology” of certain Galois-stable arithmetic subgroups of ${D}_{\phantom{\rule{0.3em}{0ex}}E}^{×}$. 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