Syntomic cohomology and p-adic regulators for varieties over p-adic fields

SYNTOMIC REGULATORS AND p-ADIC INTEGRATION I: RIGID SYNTOMIC REGULATORS

The syntomic cohomology, more precisely the cohomology of the sheaves s(n) on the syntomic site of a scheme, where introduced in [FM87] in order to prove comparison isomorphisms between crystalline and p-adic étale cohomology. It can be seen as an analogue of the Deligne-Beilinson cohomology in the p-adic world (for an excellent discussion see [Nek98]). In particular, when X is a smooth scheme ...

SYNTOMIC REGULATORS AND p-ADIC INTEGRATION II: K2 OF CURVES

CRYSTALLINE SHEAVES, SYNTOMIC COHOMOLOGY AND p-ADIC POLYLOGARITHMS

In [BD92] (see also [HW98]), A. A. Beilinson and P. Deligne constructed the motivic polylogarithmic sheaf on PQ\{0, 1,∞}. Its specializations at primitive d-th roots of unity give the Beilinson’s elements of H M(Q(μd),Q(m)) = K2m−1(Q(μd))⊗ Q (m ≥ 1), whose images under the regulator maps to Deligne cohomology are the values of m-th polylogarithmic functions at primitive d-th roots of unity. The...

p-adic Cohomology

The purpose of this paper is to survey some recent results in the theory of “p-adic cohomology”, by which we will mean several different (but related) things: the de Rham or p-adic étale cohomology of varieties over p-adic fields, or the rigid cohomology of varieties over fields of characteristic p > 0. Our goal is to update Illusie’s beautiful 1994 survey [I] by reporting on some of the many i...

Factoring Polynominals over p-Adic Fields

We give an efficient algorithm for factoring polynomials over finite algebraic extensions of the p-adic numbers. This algorithm uses ideas of Chistov’s random polynomial-time algorithm, and is suitable for practical implementation.