Equivariant torsion and base change

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 ${\Gamma }_{“E∕F}=\langle \sigma \rangle$. We prove lower bounds for “the Lefschetz number of $\sigma$ 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.