Explicit construction of K3 surfaces with everywhere good reduction

Fontaine and Abrashkin independently proved in 1985 that there are no abelian varieties defined over the rationals with good reduction at every prime. In 1991, they extended this result, and as a consequence they were able to prove that there are no K3 surfaces with everywhere good reduction over the rationals. Recently, many mathematicians have been working on how to find explicit examples of both elliptic curves and abelian surfaces with everywhere good reduction over a number field, and a natural question is then whether it is also possible to construct K3 surfaces with good reduction at every prime of a number field.

In this talk, I will explain how this can be achieved, by studying Kummer surfaces associated to the jacobians of genus two curves with everywhere good reduction. In order to do that, I will discuss how to compute the equations of the jacobian of a genus two curve in such a way that the reduction modulo 2 of the equations corresponds to the jacobian of the reduced curve, and how this can be used to study the desingularisation of Kummer surfaces over a field of characteristic two.