Trace Forms over Finite Fields of Characteristic 2 with Prescribed Invariants

AbstractWe determine the possible pairs of invariants of a trace form and construct forms with these invariants. We use this to construct new maximal Artin–Schreier curves

K3 surfaces over finite fields with given L-function

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z? Assuming semi-stable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.Comment: (v2: minor corrections, added numerical evidence by Kedlaya and Sutherland