In a commutative Noetherian local complex normed algebra which is complete in its M-adic metric there are only finitely many closed prime ideals.

André used Hodge-theoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties wit...

In these notes, we give an overview of our paper [BKT] which gives an explicit description of the de Rham and p-adic realizations of the elliptic polylogarithm, for a general elliptic curve defined over a subfield of C in the de Rham case and for a CM elliptic curve defined over its field of complex multiplication and with good reduction at the primes above p ≥ 5 in the p-adic case. As explaine...

In this paper, we investigate some properties for the (h, q)-tangent numbers and polynomials. By using these properties, we obtain some interesting identities on the (h, q)-tangent polynomials and Bernstein polynomials. Throughout this paper, let p be a fixed odd prime number. The symbol, Zp, Qp and Cp denote the ring of p-adic integers, the field of p-adic rational numbers and the completion o...

In p-adic Hodge theory there are fully faithful functors from certain categories of p-adic representations of the Galois group GK := Gal(K/K) of a p-adic field K to certain categories of semi-linear algebra structures on finite-dimensional vector spaces in characteristic 0. For example, semistable representations give rise to weakly admissible filtered (φ,N)-modules, and Fontaine conjectured th...

We determine test vectors and explicit formulas for all Bessel models for those Iwahori-spherical representations of GSp4 over a p-adic field that have non-zero vectors fixed under the Siegel congruence subgroup.

Inspired by previous work of Shoup, Lenstra-De Smit and Couveignes-Lercier, we give fast algorithms to compute in (the first levels of) the l-adic closure of a finite field. In many cases, our algorithms have quasi-linear complexity.

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.