Singular moduli for real quadratic fields: A rigid analytic approach

A rigid meromorphic cocycle is a class in the €rst cohomology of the discrete group Γ := SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid meromorphic functions on the p-adic upper half plane Hp := P1(Cp) − P1(Qp). Such a class can be evaluated at the real quadratic irrationalities in Hp, which are referred to as “RM points”. Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta li‰s: their zeroes and poles are contained in a €nite union of Γ-orbits of RM points, and their RM values are conjectured to lie in ring class €elds of real quadratic €elds. ‘ese RM values enjoy striking parallels with the CM values of modular functions on SL2(Z)\H: in particular they seem to factor just like the di‚erences 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 factorisation of the resulting singular moduli for real quadratic €elds.