Most hyperelliptic curves over Q have no rational points

By Manjul Bhargava. 2013

55 citations recorded by the source corpus.

By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/7ec85f79bbf0ddda71ed4f98da02015010c05a90

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Information Systems

Citations in the corpus

Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces (2015, unknown): “ Cohen–Lenstra–Martinet heuristics for class groups, to bound the average rank of elliptic curves over Q, and to show that most hyperelliptic curves over Qhave few rational points (see, e.g., [6, 7], [9]–[23], [35], [38], [40] for these and further examples). The representations that have so far bee”
Most odd degree hyperelliptic curves have only one rational point (2013, background): “Independently of the present paper, Bhargava [Bha13] has proved that in the family of all not-necessarily-monic even-degree genus g hyperelliptic curves over Q, the density of those that have no rational points is 1 − o(2−g) as g → ∞.”