For any prime $r \in \mathbb N$ and almost all $k smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes expected number of values, as $n$ ranges from 1 to $x$. As a consequence, deduce statements concerning var

Let F0 = Q(√ −d) be an imaginary quadratic field with 3 ∤ d and let K0 = Q(√ 3d). Let ε0 be the fundamental unit of K0 and let λ be the Iwasawa λ-invariant for the cyclotomic Z3-extension of F0. The theory of 3-adic L-functions gives conditions for λ ≥ 2 in terms of ǫ0 and the class numbers of F0 and K0. We construct units of K1, the first level of the Z3-extension of K0, that potentially occur...

We find formulas for the birational maps from a Kummer surface K and its dual K∗ to their common minimal desingularization S . We show how the nodes of K blow up. Then we give a description of the group of linear automorphisms of S . Mathematics Subject Classification (2000). Primary 14J28, 14M15; Secondary 14J50.

In this note we prove the existence of smooth Kummer surfaces in projective three-space containing sixteen mutually disjoint smooth rational curves of any given degree. Introduction Let X be a smooth quartic surface in projective three-space P. As a consequence of Nikulin’s theorem [6] X is a Kummer surface if and only if it contains sixteen mutually disjoint smooth rational curves. The classic...

In this paper, we introduce a new differential-difference operator Tξ (ξ ∈ R ) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this o...

This paper explores some foundational results of algebraic number theory. We focus on Dedekind rings and unique factorization of prime ideals, as well as some celebrated consequences such as a partial proof of Fermat’s last theorem due to Kummer, and the law of quadratic reciprocity.

In this paper we show that there is a correspondence between some K3 surfaces with non-isometric transcendental lattices constructed as a twist of the transcendental lattice of the Jacobian of a generic genus 2 curve. Moreover, we show the existence of a correspondence between a general K3 surface with ρ = 17 and a Kummer surface having transcendental lattices Q-Hodge isomorphic.

We consider powers of a grossencharacter, the corresponding Lfunctions twisted with quadratic Dirichlet characters, and their central critical values. We state several conjectures concerning Kummer-type congruences between these numbers for a ramified prime and describe specific numerical data in support of these conjectures.

In this paper the bimatrix variate beta type IV distribution is derived from independent Wishart distributed matrix variables. We explore specific properties of this distribution which is then used to derive the exact expressions of the densities of the product and ratio of two dependent Wilks’s statistics and to define the bimatrix Kummer-beta type IV distribution.

Kummer, M. and F. Stephan, Weakly semirecursive sets and r.e. orderings, Annals of Pure and Applied Logic 60 (1993) 133-150. Weakly semirecursive sets have been introduced by Jockusch and Owings (1990). In the present paper their investigation is pushed forward by utilizing r.e. partial orderings, which turn out to be instrumental for the study of degrees of subclasses of weakly semirecursive s...