Most odd degree hyperelliptic curves have only one rational point. (English) Zbl 1303.11073

Poonen, Bjorn; Stoll, Michael

Ann. Math. (2) 180, No. 3, 1137-1166 (2014).

In the present article, the authors consider hyperelliptic curves of odd degree of the form \(y^{2}=f(x)\), where \(f(x)\) is a monic irreducible polynomial, of degree \(2g+1\) and \(g \geq 3\), with integer coefficients and without multiple zeros in \({\mathbb C}\). All of these curves, after completing with the point at infinity, have the point \(\infty\) as a rational point. In some cases the curve may or not have more rational points. The main result is that there a lot of such curves with a unique rational point (this being \(\infty\)) and the fraction of curves with that property converges to \(1\) as \(g\) goes to \(\infty\). The main arguments uses \(p\)-adic analysis.

Reviewer: Ruben A. Hidalgo (Temuco)

MSC:

11G30	Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G25	Global ground fields in algebraic geometry

Keywords:

hyperelliptic curve; rational point; Selmer group

Software:

KoszulDivisorOnPic14M8

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