A quaternionic construction of $p$-adic singular moduli

TL;DR

This paper extends the theory of $p$-adic singular moduli by constructing quaternionic cohomology classes over totally real fields, providing conjectural algebraicity results and supporting numerical evidence.

Contribution

It introduces a quaternionic construction of cohomology classes replacing $ ext{SL}_2(Z[1/p])$, and explores their evaluation at elements in almost totally complex extensions, conjecturing algebraicity.

Findings

Numerical evidence supports the algebraicity conjecture.

Construction generalizes $p$-adic singular moduli to quaternionic settings.

Conjecture that values lie in algebraic extensions of the base field.

Abstract

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural $p$-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of $\mathrm{SL}_2(\mathbb{Z}[1/p])$ which can be evaluated at real quadratic irrationalities and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article we present a similar construction of cohomology casses in which $\mathrm{SL}_2(\mathbb{Z}[1/p])$ is replaced by an order in an indefinite quaternion algebra over a totally real number field $F$. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions $K$ of $F$, and we conjecture that the corresponding values lie in algebraic extensions of $K$. We also report on extensive numerical evidence for this algebraicity conjecture.