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A positive proportion of locally soluble hyperelliptic curves over \(\mathbb {Q}\) have no point over any odd degree extension. (English) Zbl 1385.11043

Let \(C\) be a hyperelliptic curve over \(\mathbb{Q}\) of genus \(g\), expressed by an equation of the form \(z^2 = f(x,y)\) where \(f(x,y)\) is a homogeneuous polynomial of degree \(n=2g+2\) with integer coefficients \(a_0,a_1,\dots,a_n\), and \(f\) factors into distinct linear factors over \(\overline{\mathbb{Q}}\), and let the height of \(C\) be \(H(C):=\max\{ | a_k| : k = 0,\dots,n \}\). Given a positive integer \(X\) there are finitely many curves \(C\) with \(H(C)\leq X\), and the results of this paper use the enumeration determined by the finiteness of the number of curves with bounded height.
If \(C\) is not soluble, it does not have a rational point, and a harder question to answer is the rational points of \(C\) that are locally soluble. It is known [B. Poonen and M. Stoll, in: Topics in number theory. In honor of B. Gordon and S. Chowla. Proceedings of the conference, Pennsylvania State University, University Park, PA, USA, July 31–August 3, 1997. Dordrecht: Kluwer Academic Publishers. 241–244 (1999; Zbl 1024.11047)] that for fixed genus \(g\) more than 75% of the curves are locally soluble. In this paper, the authors prove that given genus \(g\) a positive proportion of locally soluble hyperelliptic curves \(C\) of genus \(g\) contain no rational points over any field extensions of \(\mathbb{Q}\) of odd degree \(\leq m\), and that given a field extension degree \(m\) a proportion of hyperelliptic curves \(C\) of genus \(g\) over \(\mathbb{Q}\) containing no points over all extensions of \(\mathbb{Q}\) of odd degree \(\leq m\) approaches \(1\) as \(g\to \infty\).
The authors’ approach is focused around the principal homogeneous space \(J^1=\mathrm{Pic}^1_{C/\mathbb{Q}}\) for \(J\). A point on \(C\) defined over an extension field of \(\mathbb{Q}\) of odd degree gives a rational point on \(J^1\), and they prove that there is a positive proportion of locally soluble hyperelliptic curves such that a certain Selmer set \(\mathrm{Sel}_2(J^1)\) is empty, which in turn implies that \(C \) has no rational points. The authors also introduce a similar statistical result on the Brauer-Manin obstructions to \(J^1\) having a rational point.

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points

Citations:

Zbl 1024.11047
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References:

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