On the syntomic regulator for products of elliptic curves

Copyright © 2011 London Mathematical SocietyWe consider the syntomic regulator on the integral motivic cohomology of a smooth proper surface over a p-adic field and apply a recent formula of Besser that uses p-adic integration theory, in particular his theory of triple indices on Coleman integrals, to the case of a self-product of an elliptic curve. The method is suitable to separate decomposable from indecomposable elements in the (integral) motivic cohomology. As an interesting example, we construct an element that, though not given in decomposable form, becomes decomposable after taking p-adic completion

p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.Comment: 40 page

The rigid syntomic ring spectrum

The aim of this paper is to show that Besser syntomic cohomology is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induces a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of syntomic coefficients.Comment: Final version to appear in the Journal de l'institut des Math\'ematiques de Jussieu. Many typos have been corrected and the exposition has been improved according to the suggestions of the referees: we thank them a lot

New p-adic hypergeometric functions and syntomic regulators

We introduce new p-adic convergent functions, which we call the p-adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork's. The second main result is that the special values of our new functions appear in the syntomic regulators for hypergeometric curves, Fermat curves and some elliptic curves. According to the p-adic Beilinson conjecture by Perrin-Riou, they are expected to be related with the special values of p-adic L-functions. We provide one example for this.Comment: 54 pages, totally revised versio

Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series

This article is the first in a series devoted to the Euler system arising from p-adic families of Beilinson-Flach elements in the first K-group of the product of two modular curves. It relates the image of these elements under the p-adic syntomic regulator (as described by Besser (2012)) to the special values at the near-central point of Hida's p-adic Rankin L-function attached to two Hida families of cusp forms.Peer ReviewedPostprint (author’s final draft