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2016 K3 surfaces over finite fields with given $L$-function
Lenny Taelman
Algebra Number Theory 10(5): 1133-1146 (2016). DOI: 10.2140/ant.2016.10.1133

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 (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.

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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

Information

Received: 17 August 2015; Revised: 27 November 2015; Accepted: 27 December 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1344.14024
MathSciNet: MR3531364
Digital Object Identifier: 10.2140/ant.2016.10.1133

Subjects:
Primary: 14J28
Secondary: 11G25, 14G15, 14K22

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.10 • No. 5 • 2016
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