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Publication# Shimura Curves and Special Values of p-adic L-functions

Abstract

We construct "generalized Heegner cycles" on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-function L-p(f, chi) which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series theta(chi) attached to an unramified Hecke character of an imaginary quadratic field K. This generalizes previous work of Bertolini, Darmon, and Prasanna, which demonstrated a similar result in the case of modular curves. Our main tool is the theory of Serre-Tate coordinates, which yields p-adic expansions of modular forms at CM points, replacing the role of q-expansions in computations on modular curves.

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Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf{Q}, the rational numbers.
Every such quadratic field is some \mathbf{Q}(\sqrt

Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies:

- a kind of functional equation with respect to the group action of the modular group,
- and

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers \mathbb{Q} such that

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.