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arXiv · 2010.06898

A quaternionic construction of $p$-adic singular moduli

Xavier Guitart·Marc Masdeu·Xavier Xarles
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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.

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Xavier Guitart, Marc Masdeu, Xavier Xarles. 2020-10-14. A quaternionic construction of $p$-adic singular moduli. https://arxiv.org/abs/2010.06898

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