A quaternionic construction of p-adic singular moduli

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 construction of cohomology classes inspired by that of Darmon–Vonk, 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.