The Universal Kummer Threefold

The Universal Kummer Threefold

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...

The Universal Kummer Congruences

Let p be a prime. In this paper, we present detailed p-adic analysis to factorials and double factorials and their congruences. We give good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p−1. Using these we then establish the universal Kummer congruences modulo powers of a prime p for the divided universal Bernoulli numbers...

Universal Kummer Congruences Mod Prime Powers

Bounds of Divided Universal Bernoulli Numbers and Universal Kummer Congruences

Let p be a prime. We obtain good bounds for the p-adic sizes of the coefficients of the divided universal Bernoulli number B̂n n when n is divisible by p− 1. As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo p for the divided universal Bernoulli numbers for the case (p − 1)|n, which is a new result.

Kummer Coverings and Specialization

We generalize various classical results on specialization of fundamental groups to log schemes in the sense of Fontaine and Illusie. The key technical result relates the category of finite Kummer étale covers of a fs log scheme over a complete noetherian local ring to the Kummer étale coverings of its reduction, generalizing earlier results of Hoshi.