On quaternionic rigid meromorphic cocyles

TL;DR

This paper generalizes the algebraicity of divisors associated with rigid meromorphic cocycles from $ ext{SL}_2(bZ[1/p])$ to $rakp$-arithmetic subgroups over arbitrary number fields, using Bieri-Eckmann duality.

Contribution

It extends the theory of rigid meromorphic cocycles to more general arithmetic groups and introduces a new proof technique based on Bieri-Eckmann duality.

Findings

Proved algebraicity of divisors in a broader setting.

Developed a new proof method avoiding modular symbols.

Generalized previous results to arbitrary number fields.

Abstract

Recently, Darmon and Vonk initiated the theory of rigid meromorphic cocycles for the group $\mathrm{SL}_2(\mathbb{Z}[1/p])$. One of their major results is the algebraicity of the divisor associated to such a cocycle. We generalize the result to the setting of $\mathfrak{p}$-arithmetic subgroups of inner forms of $\mathrm{SL}_2$ over arbitrary number fields. The method of proof differs from the one of Darmon and Vonk. Their proof relies on an explicit description of the cohomology via modular symbols and continued fractions, whereas our main tool is Bieri-Eckmann duality for arithmetic groups.