On quaternionic rigid meromorphic cocyles

Recently, Darmon and Vonk initiated the theory of rigid meromorphic cocycles for the group $\mathbb{SL}_2 (\mathbb{Z}[1/p])$. 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 $\mathbb{SL}_2$ 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.

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Received 10 September 2020

Accepted 23 February 2021

Published 21 April 2023