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# Zeta Functions of Supersingular Curves of Genus 2

Published:2007-04-01
Printed: Apr 2007
• Daniel Maisner
• Enric Nart
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## Abstract

We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.
 MSC Classifications: 11G20 - Curves over finite and local fields [See also 14H25] 14G15 - Finite ground fields 11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]