Abstract
The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and -adic) and a number of less obvious (-adic) constraints. We consider the converse question, in the style of Honda–Tate: given a function satisfying all these constraints, does there exist a K3 surface whose zeta-function equals ? 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.
Citation
Lenny Taelman. "K3 surfaces over finite fields with given $L$-function." Algebra Number Theory 10 (5) 1133 - 1146, 2016. https://doi.org/10.2140/ant.2016.10.1133
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