15 January 2021 Singular moduli for real quadratic fields: A rigid analytic approach
Henri Darmon, Jan Vonk
Duke Math. J. 170(1): 23-93 (15 January 2021). DOI: 10.1215/00127094-2020-0035


A rigid meromorphic cocycle is a class in the first cohomology of the discrete group Γ : = SL 2 ( Z [ 1 / p ] ) with values in the multiplicative group of nonzero rigid meromorphic functions on the p -adic upper half-plane H p : = P 1 ( C p ) P 1 ( Q p ) . Such a class can be evaluated at the real quadratic irrationalities in H p , which are referred to as “RM points.” Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lifts: their zeroes and poles are contained in a finite union of Γ -orbits of RM points, and their RM values are conjectured to lie in ring class fields of real quadratic fields. These RM values enjoy striking parallels with the values of modular functions on SL 2 ( Z ) \ H at complex multiplication (CM) points: in particular, they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p -adic accuracy leads to convincing numerical evidence for the algebraicity and factorization of the resulting singular moduli for real quadratic fields.


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Henri Darmon. Jan Vonk. "Singular moduli for real quadratic fields: A rigid analytic approach." Duke Math. J. 170 (1) 23 - 93, 15 January 2021. https://doi.org/10.1215/00127094-2020-0035


Received: 22 December 2018; Revised: 24 March 2020; Published: 15 January 2021
First available in Project Euclid: 5 January 2021

MathSciNet: MR4194897
Digital Object Identifier: 10.1215/00127094-2020-0035

Primary: 11R37
Secondary: 11G15

Keywords: Complex Multiplication , explicit class field theory , modular geodesics , rigid meromorphic cocycles

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 1 • 15 January 2021
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