The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble’s quartic hypersu...

In the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm (QPA) is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular,...

Let p be a prime and Fq be a finite field with q = p m elements and let F̄q be an algebraic closure of Fq . In this paper we present a method for constructing curves over finite fields with many points which are Kummer covers of P, or of other suitable base curves. For this we look at rational functions f ∈ Fq(x) such that f assumes a fixed value a ∈ Fq on a (preferably large) set P ⊆ P(Fq). To ...

We show that every supersingular K3 surface in characteristic 2 with Artin invariant ≤ 2 is obtained by the Kummer type construction of Schröer.

In this note we give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger’s theorem.