an:06399397
Zbl 1387.14086
Br??ker, Reinier; Howe, Everett W.; Lauter, Kristin E.; Stevenhagen, Peter
Genus-2 curves and Jacobians with a given number of points
EN
LMS J. Comput. Math. 18, 170-197 (2015).
00340627
2015
j
14H45 14K22 11G15 11G20 14G15 14H40
genus; curves; Jacobian
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)], \textit{R. Br??ker} and \textit{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.
Lei Zhang (Berlin)
Zbl 1127.14022