The purpose of the present paper is to show that morphisms between the generic fibers of truncated Barsotti-Tate group schemes over mixed characteristic complete discrete valuation rings with perfect residue fields extend in a “tame-blind” fashion — i.e., under a condition which is unaffected by passing to a tame extension — to morphisms between the original truncated Barsotti-Tate group scheme...

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

1.1. The p-adic measure. Let p be a prime number. We consider a field K with a valuation ord : K× → Z, extended to K by ord(0) = ∞. We denote by OK the valuation ring OK = {x ∈ K|ord(x) ≥ 0} and we fix an uniformizing parameter $, that is, an element of valuation 1 in OK . The ring OK is a local ring with maximal ideal MK of OK generated by $. We shall assume the residue field k := OK/MK is fin...

In this paper we consider the Newton polygons of L-functions coming from additive exponential sums associated to a polynomial over a finite field Fq. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree d ≥ 2 when the characteristic p is greater than 3d, and the Hasse polyn...

Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F ) that picks out a G(F )-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F ). This generalizes an earlier result obtained by DeBack...

1.1. The p-adic numbers. A rational number x ∈ Q× may be uniquely written as x = ab p n with a, b and n nonzero integers such that p ab. We define ordp(x) = n, |x|p = p−n, |0|p = 0. |·|p defines an absolute value on Q, satisfying the stronger ultrametric triangle equality |x+ y|p ≤ max(|x|p, |y|p). We define Qp to be the completion Q with respect to this metric and we use the same notation | · ...

We analyze properties of the 2-adic valuations of S(n, k), the Stirling numbers of the second kind. A conjecture that describes patterns of these valuations for fixed k and n modulo powers of 2 is presented. The conjecture is established for k = 5.

The p-typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields k of prime characteristic p > 0 to p-adically complete discrete valuation rings of characteristic 0 with residue field k and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor WG attached to...