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, 14G15, 11G10 show english descriptions Curves over finite and local fields [See also 14H25]
Finite ground fields
Abelian varieties of dimension $> 1$ [See also 14Kxx] 11G20 - Curves over finite and local fields [See also 14H25]
14G15 - Finite ground fields
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]

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