The p-adic upper half plane

The p-adic upper half plane X is a rigid analytic variety over a p-adic field K, on which the group GL2(K) acts, that Mumford introduced (as a formal scheme) as part of his efforts to generalize Tate’s p-adic uniformization of elliptic curves to curves of higher genus. The Cp–valued points of X are just P(Cp)−P(K), with GL2(K) acting by linear fractional transformations. Mumford showed that the appropriate generalization of Tate’s elliptic curves – the “totally split” curves of higher genus – could be constructed as the quotient of the space X by appropriate discrete groups Γ ∈ GL2(K). Mumford’s work acquired even greater significance for number theorists when Cerednik and Drinfeld showed that an important class of modular curves – the Shimura curves – could be constructed by p-adic uniformization by choosing for the discrete group Γ an appropriate arithmetic subgroup coming from a definite quaternion algebra over Q. More recently, the p-adic upper half plane has figured prominently in recent developments in arithmetic geometry. In Section 1 of these notes, we will construct the space X as a rigid variety and describe some of its most fundamental geometric properties, and in subsequent sections we will explore some of this more recent work. Our focus in Section 2 will be the analytic theory of X, and in particular the relationship between spaces of functions on the p-adic upper half plane and distributions on P(K), which is the “boundary” of X. One main result will be the construction of the Poisson integral for X; in a manner analogous to the classical Poisson transform, this integral allows one to recover rigid analytic functions on X from appropriate boundary distributions by integrating against a kernel function. In Sections 3 and 4, we establish connections between number theory and the geometry of the p-adic upper half plane, with particular emphasis on the relationship between the p-adic upper half plane and L-invariants. If E is an elliptic curve over Q with split multiplicative reduction at p and analytic Mordell-Weil rank zero, then [25] conjectured and [19] proved that the p-adic L-function of the modular form f corresponding to E vanishes to order 1 and the special value of Lp(1) differs from the classical special value by the number