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Genus-2 curves and Jacobians with a given number of points. (English) Zbl 1387.14086
Given an algebraic variety \(V\) over a finite field \(\mathbb{F}_q\), there are only finitely many rational points on \(V\) as \(V\) is of finite type. So it becomes a problem of counting the number of rational points for such varieties. The paper under review considers the converse of this problem:
Given a positive integer \(N\), how to construct a finite field \(\mathbb{F}_q\) and smooth varieties \(V\) such that the set of rational points \(V(\mathbb{F}_q)\) has cardinality \(N\).
In [Math. Comput. 76, No. 260, 2161–2179 (2007; Zbl 1127.14022)], R. Bröker and P. Stevenhagen deal with the problem for elliptic curves. In this paper the authors generalize the result of [loc. cit.] to curves of genus 2.
For a smooth projective connected curve \(C\) over a field \(k\), one can define an abelian variety \(J(C)\) which is isomorphic to \(\text{Pic}_{C/k}^0\). Choosing a point in \(C(k)\), one can imbed \(C\) into \(J(C)\), and this imbedding is an isomorphism if \(C\) is an elliptic curve. This leads to two generalizations of the problem to genus 2 case:
\(\bullet\) Construct a finite field \(\mathbb{F}_q\) and curves of genus 2 with \(N\) \(\mathbb{F}_q\)-rational points;
\(\bullet\) Construct a finite field \(\mathbb{F}_q\) and curves of genus 2 whose Jacobian has \(N\) \(\mathbb{F}_q\)-rational points;
In this paper the authors consider both of the generalisations.
Reviewer: Lei Zhang (Berlin)

14H45 Special algebraic curves and curves of low genus
14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
11G20 Curves over finite and local fields
14G15 Finite ground fields in algebraic geometry
14H40 Jacobians, Prym varieties
genus; curves; Jacobian
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