Evaluation of Bianchi Rigid Meromorphic Cocycles at Big ATR Points

TL;DR

This paper develops computational tools to evaluate Bianchi rigid meromorphic cocycles at big ATR points, providing the first numerical evidence for their algebraicity and revealing behavior similar to Borcherds products at big CM points.

Contribution

It introduces methods for evaluating Bianchi cocycles at big ATR points and verifies their algebraicity numerically for the first time.

Findings

Numerical verification of algebraicity of special values.

Behavior analogous to Borcherds products at big CM points.

First computational evidence supporting conjectures.

Abstract

We develop the tools required to effectively evaluate the Bianchi rigid meromorphic cocycles introduced by Darmon-Gehrmann-Lipnowski at big ATR points, and use them to obtain the first numerical verification of the conjectured algebraicity of these special values. Moreover, our computations suggest that these special values exhibit behaviour analogous to that of the special values of Borcherds products on Hilbert modular surfaces at big CM points.