The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point. (English) Zbl 1303.11072

Bhargava, Manjul; Gross, Benedict H.

Prasad, D. (ed.) et al., Automorphic representations and \(L\)-functions. Proceedings of the international colloquium, Mumbai, India, January 3–11, 2012. New Delhi: Hindustan Book Agency; Mumbai: Tata Institute of Fundamental Research (ISBN 978-93-80250-49-6/hbk). Studies in Mathematics. Tata Institute of Fundamental Research 22, 23-91 (2013).

Summary: We prove that when all hyperelliptic curves of genus \(n \geq 1\) having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2.
The method of C. Chabauty [C. R. Acad. Sci., Paris 212, 882–885 (1941; Zbl 0025.24902; JFM 67.0105.01)] can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus \(n \geq 3\) with a rational Weierstrass point have fewer than 20 rational points.
For the entire collection see [Zbl 1293.11002].

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G05	Elliptic curves over global fields
14H55	Riemann surfaces; Weierstrass points; gap sequences
14H40	Jacobians, Prym varieties

Keywords:

Selmer group; Jacobian; hyperelliptic curves; Weierstrass point; Mordell-Weil group

Citations:

Zbl 0025.24902; JFM 67.0105.01