A quaternionic construction of p-adic singular moduli

Xavier Guitart*, Marc Masdeu, Xavier Xarles

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)

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 SL2(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 SL2(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.

Original languageEnglish
Article number45
JournalResearch in Mathematical Sciences
Volume8
Issue number3
DOIs
Publication statusPublished - Sep 2021

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