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

Published online by Cambridge University Press:  26 August 2016

Kiran S. Kedlaya
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093, USA email [email protected]
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email [email protected]

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.

Type
Research Article
Copyright
© The Author(s) 2016 

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