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A census of zeta functions of quartic K $3$ surfaces over $\mathbb{F}_{2}$

Part of: Surfaces and higher-dimensional varieties Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 August 2016

Kiran S. Kedlaya  and
Andrew V. Sutherland
Kiran S. Kedlaya
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093, USA email kedlaya@ucsd.edu
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email drew@math.mit.edu

Abstract

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We compute the complete set of candidates for the zeta function of a K $3$ surface over $\mathbb{F}_{2}$ consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over $\mathbb{F}_{2}$ . These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K $3$ surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

MSC classification

Primary: 11M38: Zeta and $L$-functions in characteristic $p$ 14J28: $K3$ surfaces and Enriques surfaces

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