p-adic Eichler-Shimura maps for the modular curve

2006

2014

This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman–de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur–Swinnerton–Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.

Duke Mathematical Journal, 1999

2012

Abstract. This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br] and

Beilinson's Conjectures on Special Values of L-Functions, 1988

Contents 0. Introduction. 1. The Theorem. 2. Transformation of L-values. 3. Eisenstein series and modular units. 4. Whittaker functions and L-factors. 5. Evaluation of the regulator integral. 6. Non-vanishing of the regulator integral. 7. Integrality. References.

arXiv (Cornell University), 2020

We prove p-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew hermitian spaces V with respect to an arbitrary CM field K with maximal totally real subfield F . For a place v|p of F that is not split in K and for which Vv is anisotropic, let ν be an extension of v to the reflex field E. We define an integral model of the corresponding Shimura curve over Spec O E,(ν) by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to p. The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane Ω Fv for Fv. The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal O Fv -modules with a Rapoport-Zink space of p-divisible groups which arise from the moduli problem, where the O Fv -action is usually not strict when Fv = Qp. Our main tool is the theory of displays, in particular the Ahsendorf functor.

Inventiones Mathematicae, 2000

Inventiones Mathematicae, 1993

2006

Proposition 1.0.2. i.e., φ 1 is the q-expansion of E 1 (τ, 1 , B). 2 pending on B. 1 Alternatively, φ1(τ ) can be obtained by calculating the integral over M(C) of a theta function valued in (1, 1) forms; this amounts to a very special case of the results of . The analogous computation in the case of modular curves was done by Funke .

… Unlimited–2001 and Beyond, Engquist, Schmid, …, 2001