2013/08/02 by Manjul Bhargava, Bhargava, Manjul · 5 citations
Computer Science · Mathematics · #11G30 #14H25 #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1308.0395
openalex publication_date 2013/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 P1 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 : z2 = f(x,y) = f0 xn + f1 xn-1 y + ... + fn yn where n=2g+2, the coefficients fi lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max|fi|, 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.