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A quaternionic construction of p-adic singular moduli
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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 DarmonVonk, 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.
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GUITART MORALES, Xavier, MASDEU, Marc and XARLES RIBAS, Francesc Xavier. A quaternionic construction of p-adic singular moduli. Research in the Mathematical Sciences. 2021. Vol. 8. ISSN 2522-0144. [consulted: 4 of August of 2026]. Available at: https://hdl.handle.net/2445/194821