Holomorphic torsion and geometric zeta functions for certain Hermitian locally symmetric manifolds

TL;DR

This paper connects holomorphic torsion of certain Hermitian locally symmetric manifolds to a Weil-type zeta function, revealing geometric and dynamical insights, especially for rank one and two groups.

Contribution

It introduces a new zeta function description of holomorphic torsion for specific Hermitian locally symmetric manifolds, incorporating their compactification geometry.

Findings

Zeta function construction involves the geometry of compactifications.

For real rank one groups, the Ansatz holds for all coefficients.

For some rank two groups, the Ansatz holds for certain coefficients.

Abstract

We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a technical Ansatz relative to the given coefficients. A distinguishing feature of our zeta function is that its construction involves in an essential way the geometry of a standard compactification of the universal cover. The two senior authors are indebted to their junior colleague, Jan Frahm, for his laborious work shedding light on the scope of the validity of the Ansatz, and for writing up the attached Appendix. The results therein show that for real rank one groups G the Ansatz is satisfied with respect to any coefficients, for some rank two groups G it is satisfied with respect to certain coefficients, and also that there are groups G which do not obey the Ansatz.