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On the non-existence of certain curves of genus two. (English) Zbl 1067.11035
Let \(q\) be a power of an odd prime and \(f(x)=f_q(x):=x^4+(2-2q)x+q^2\). For \(q\leq 61\) D. Maisner and E. Nart [Exp. Math. 11, 321–337 (2002)] noticed that there is no curve of genus 2 over \(F_q\) whose characteristic polynomial is \(f(x)\). The objective of this paper is to show that such statement holds for any \(q\). The prove is based on a counting argument. First the number of principally polarized abelian surfaces with characteristic polynomial \(f(x)\) is counted. It turns out that this number is equal to the number of geometrically split principally polarized abelian surfaces whose characteristic polynomial is \(f(x)\). In particular, \(f(x)\) cannot be the characteristic polynomial of a Jacobian. It turns out that for an abelian surface with characteristic polynomial \(f(x)\), \((End A)\otimes \mathbb Q=K\) where \(K=\mathbb Q(\sqrt{-2},\sqrt{-D})\), \(2q-1=F^2D\) with \(D\) squarefree. Hence the approach of proof of the main result in this paper is based on several properties of the subfields of \(K\).

MSC:
11G20 Curves over finite and local fields
11G10 Abelian varieties of dimension \(> 1\)
11R65 Class groups and Picard groups of orders
14G15 Finite ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
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