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

Let OK be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. From the semi-stable conjecture (Cst) and the theory of slopes, we obtain isomorphisms between the maximal unramified quotients of certain Tate twists of p-adic étale cohomology groups and the cohomology groups of logarithmic Hodge-Witt sheaves for a proper semi-stable scheme over OK . The ob...

Consider the following setup: let G be an algebraic group over some field k. What can we say about finite subgroups of G? In particular, is it possible to bound above the orders of such subgroups? The history of such a question goes a long way back, starting from Minkowski, then Schur and others. We will briefly introduce some weak bounds given by Minkowski, then proceed to give bounds which ar...

The aim of this paper is to generalize the‎ ‎notion of almost valuation domains to arbitrary commutative‎ ‎rings‎. ‎Also‎, ‎we consider relations between almost valuation rings ‎and pseudo-almost valuation rings‎. ‎We prove that the class of‎ ‎almost valuation rings is properly contained in the class of‎ ‎pseudo-almost valuation rings‎. ‎Among the properties of almost‎ ‎valuation rings‎, ‎we sh...

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f ∈ Int(D), we explicitly construct a divisor homomorphism from [[f ]], the divisor-closed submonoid of Int(D) generated by f , to a finite sum of copies of (N0,+). This implies that [[f ]] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V ...

Abstract A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x) ∈ Qp[x], we consider the sequence {vp(f(n))}n=0, where vp is the p-adic valuation. We show that this sequence...

The weight-monodromy conjecture claims the coincidence of the shifted weight filtration and the monodromy filtration on étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it was already proved in some cases, the case of dimension ≥ 3 in mixed characteristic is still unproved up to now. The aim of this paper is to prove the weight-monodromy conjecture ...

If R is a Dedekind domain, P a prime ideal of R and S ⊆R a finite subset then a P -ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101–127, 1997), is an ordering {ai}i=1 of the elements of S with the property that, for each 1 < i ≤m, the choice of ai minimizes the P -adic valuation of ∏j<i(s− aj ) over elements s ∈ S. If S, S′ are two finite subsets of R of the same ca...

Let k be a locally compact non-discrete field with non-Archimedean valuation (Say, just the p-adic numbers Qp), O its ring of integers (say Zp) and P a generator for the maximal ideal of O (i.e. p). In future lectures we will see how Hecke algebras relate to the representation theory of reductive algebraic groups over these fields. The main example to keep in mind is G = GL(n, k). When I talk a...

For a polynomial f(x) in (Zp∩Q)[x] of degree d ≥ 3 let L(f⊗Fp;T ) be the L function of the exponential sum of f mod p. Let NP(f ⊗ Fp) denote the Newton polygon of L(f⊗Fp; T ). Let HP(f) denote the Hodge polygon of f , which is the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d−1. We prove that there is a Zariski dense subset U defined over Q in the space A of degree-d monic...