Singular Moduli for Real Quadratic Fields: a Rigid Analytic Approach

A rigid meromorphic cocycle is a class in the first 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 planeHp := P1(Cp)−P1(Qp). Such a class can be evaluated at the real quadratic irrationalities in Hp, which are referred to as “RM points”. The RM values of arbitrary rigid meromorphic cocycles (whose restriction to a parabolic subgroup take values in Cp ) are conjectured to lie in ring class fields of real quadratic fields and to enjoy striking parallels with the CM values of modular functions on SL2(Z)\H: in particular they seem to factor just like the differences of classical singular moduli, as described by Gross and Zagier in [GZ1]. A fast algorithm for computing rigid meromorphic cocycles to high padic accuracy leads to convincing numerical evidence for the algebraicity and factorisation of the resulting singular moduli for real quadratic fields.