THE $p$-ADIC UNIFORMISATION OF MODULAR CURVES BY $p$-ARITHMETIC GROUPS (Profinite monodromy, Galois representations, and Complex functions)

Abstract

This is a transcription of the author's lecture at the Kyoto conference "Profinite monodromy, Galois representations, and complex functions" marking Yasutaka Ihara's 80th birthday. Much of it, notably the material in the last section, is the fruit of an ongoing collaboration with Jan Vonk. In his important work on "congruence monodromy problems", Professor Ihara proposed that the group Gamma :=SL_{2}(mathbb{Z}[1/p]) acting on the product of a Drinfeld and a Poincaré upper half-plane provides a congenial framework for describing the ordinary locus of the j-line in characteristic p. In Ihara's picture, which rests on Deuring's theory of the canonical lift, the ordinary points of the j-line are essentially in bijection with conjugacy classes in Gamma that are hyperbolic at p and elliptic at infty. The present note explains how the classes that are elliptic at p and hyperbolic at infty form the natural domain for a kind of p-adic uniformisation of the modular curve X_{0}(p), leading to a conjectural analogues of Heegner points, elliptic units, and singular moduli defined over ring class fields of real quadratic fields.