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.
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Notes
i.e., when p belongs to the set \(\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 \}\)
we will assume that \(c\ne 0\), as this can always be achieved by replacing \(\psi \) by a conjugate if necessary
Available at https://github.com/mmasdeu/darmonvonk
Maintained by the second named author at https://github.com/mmasdeu/darmonpoints
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Acknowledgements
We thank Jan Vonk and Henri Darmon for illuminating conversations about their work and for sharing with us the code used in their original calculations. We also thank them for suggesting to us the interest of the numerical verification that we present in Sect. 8.3. Guitart was partially funded by project PID2019-107297GB-I00. Xarles was partially supported by project MTM2016-75980-P. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 682152).
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Guitart, X., Masdeu, M. & Xarles, X. A quaternionic construction of p-adic singular moduli. Res Math Sci 8, 45 (2021). https://doi.org/10.1007/s40687-021-00274-3
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DOI: https://doi.org/10.1007/s40687-021-00274-3