Singular moduli for real quadratic fields: A rigid analytic approach

Abstract

A rigid meromorphic cocycle is a class in the first cohomology of the discrete group $\Gamma :={\mathrm{SL}}_2\left(\mathbb{Z}\right[1/p\left]\right)$ with values in the multiplicative group of nonzero rigid meromorphic functions on the $p$-adic upper half-plane ${\mathcal{H}}_p:={\mathbb{P}}_1\left({\mathbb{C}}_p\right)-{\mathbb{P}}_1\left({\mathbb{Q}}_p\right)$. Such a class can be evaluated at the real quadratic irrationalities in ${\mathcal{H}}_p$, 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 $\Gamma$-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 ${\mathrm{SL}}_2\left(\mathbb{Z}\right)\backslash \mathcal{H}$ 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.