Height pairings on Shimura curves and p-adic uniformization

Journal of Differential Geometry, 1998

In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product of Drinfeld upper half-spaces and their equivariant coverings. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [38] and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety. Moreover, we show that difficult theorems of Yau and Kottwitz appearing in [38] may be avoided.

arXiv (Cornell University), 2021

We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via BGG methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was already used in the original proof. Then, we construct overconvergent Eichler-Shimura (ES) maps for the modular curves providing "the second half" of the overconvergent ES map of Andreatta-Iovita-Stevens. We use higher Coleman theory on the modular curve developed by Boxer-Pilloni to show that the small slope part of the ES maps interpolates the classical p-adic Eichler-Shimura decompositions. Finally, we prove that the overconvergent ES maps are compatible with Poincaré and Serre pairings.

Arxiv preprint math/0204361, 2002

Abstract: Using the p-adic uniformization of Shimura varieties we determine, for some of them, over which local fields they have rational points. Using this we show in some new curve cases that the jacobians are even in the sense of Poonen and Stoll.

We construct the Eisenstein measure in several variables on a quasi-split unitary group, as a first step towards the construction of p-adic L-functions of families of ordinary holomorphic modular forms on unitary groups. The construction is a direct generalization of Katz' construction of p-adic L-functions for CM fields, and is based on the theory of p-adic modular forms on unitary Shimura varieties developed by Hida, and on the explicit calculation of non-degenerate Fourier coefficients of Eisenstein series.

Inventiones Mathematicae, 1998

Cambridge University Press eBooks, 2007

2006

American Journal of Mathematics, 2012

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over Q at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem ofČerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on elliptic curves over Q unconditional.

Compositio Mathematica, 2004

We prove a relation between a generating series for the heights of Heegner cycles on the arithmetic surface associated with a Shimura curve and the second term in the Laurent expansion at s = 1 2 of an Eisenstein series of weight 3 2 for SL(2). On the geometric side, a typical coefficient of the generating series involves the Faltings heights of abelian surfaces isogenous to a product of CM elliptic curves, an archimedean contribution, and contributions from vertical components in the fibers of bad reduction. On the analytic side, these terms arise via the derivatives of local Whittaker functions. It should be noted that s = 1 2 is not the central point for the functional equation of the Eisenstein series in question. Moreover, the first term of the Laurent expansion at s = 1 2 coincides with the generating function for the degrees of the Heegner cycles on the generic fiber and, in particular, does not vanish. Introduction Part I. Arithmetic geometry 1 The moduli stack M 2 Uniformization 3 The Hodge bundle 4 The arithmetic Picard group and the arithmetic Chow group 5 Special cycles and the generating function Part II. Eisenstein series 6 Eisenstein series of weight 3 2 7 The main identities 8 Fourier expansions and derivatives Part III. Computations: geometric 9 The geometry of Z(m) 10 Contributions of horizontal components 11 Contributions of vertical components Appendix A. The case p = 2 12 Archimedean contributions Part IV. Computations: analytic 13 Local Whittaker functions: the non-archimedean case 14 Local Whittaker functions: the archimedean case 15 The functional equation