Vol. 10, No. 5, 2016

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K3 surfaces over finite fields with given $L$-function

Lenny Taelman

Vol. 10 (2016), No. 5, 1133–1146
Abstract

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and $\ell$-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 semistable 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.

Keywords
K3 surfaces, zeta functions, finite fields
Mathematical Subject Classification 2010
Primary: 14J28
Secondary: 14G15, 14K22, 11G25