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.

刚性亚纯共环是由 Darmon 和 Vonk 引入的，作为奇异模理论到实二次基场的推测p- adic 扩展。它们是\({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\) 的某些上同调类，可以在实二次无理数下进行评估，由此获得的值被推测存在于基域的代数扩展中。在本文中，我们提出了一种受 Darmon-Vonk 启发的上同调类的构造，其中\({{\,\mathrm{SL}\,}}_2(\mathbb {Z}[1/p])\ )由全实数域F 上的不定四元数代数中的阶代替. 这些四元数上同调类可以在F 的几乎完全复杂的扩展K 中的元素处进行评估，我们推测相应的值位于K 的代数扩展中。我们还报告了这个代数猜想的大量数值证据。