Average size of 2-Selmer groups of Jacobians of odd hyperelliptic curves over function fields
In this paper, the average size of 2-Selmer groups of Jacobians of hyperelliptic curves in the function fields setting is studied. Let \(K\) be the function field \(\mathbb{F}_q(C)\), where \(C\) is a smooth curve. If \(q > 4n(2n+1)\) and \(p = \operatorname{char}\mathbb{F}_q > 2(2n-1)\), then it is proved that when all hyperelliptic curves over \(K\) with a marked Weierstrass point are ordered by height, the average size of 2-Selmer groups of their Jacobians is bounded above by \(3+ f (q)\), where \(\lim_{q\rightarrow \infty} f (q) = 0\). Furthermore, if \(\operatorname{char}\mathbb{F}_q > 2(2n-1)\), then it is proved that the average size of 2-Selmer groups of the Jacobians of hyperelliptic curves with a marked Weierstrass point and square-free discriminant is 3. The result is obtained by a geometric method which is a generalization of the one that was used to obtain the average size of 2-Selmer groups of elliptic curves [\textit{Q. P. Hố} et al., Math. Res. Lett. 21, No. 6, 1305--1339 (2014; Zbl 1316.14055)].
scientific article; zbMATH DE number 7612796
scientific article