2018/04/16 by Levent Alpöge, Alpoge, Levent
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1804.05859
openalex publication_date 2018/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that, when genus two curves C/ℚ with a marked Weierstass point are ordered by height, the average number of rational points #|C(ℚ)| is bounded. The argument follows the same ideas as the sphere-packing proof of boundedness of the average number of integral points on (quasiminimal Weierstrass models of) elliptic curves. That is, we bound the number of small-height points by hand, the number of medium-height points by establishing an explicit Mumford gap principle and using the theorem of Kabatiansky-Levenshtein on spherical codes (this technique goes back to work of Silverman, Helfgott, and Helfgott-Venkatesh), and the number of large-height points by using Bombieri-Vojta's proof of Faltings' theorem. Explicitly, in dealing with non-small-height points we prove that the number of rational points (x,y) on Cf: y2 = f(x) satisfying h(x) > 8 h(f) is ≪ 1.872rank(Jac(C)(ℚ)), which has finite average by the theorem of Bhargava-Gross on the average size of 2-Selmer groups of Jacobians over this family. We note that our arguments in the small-height and large-height cases extend to general genera g≥ 2, though for medium points we need to use Stoll's bounds on the non-Archimedean local height differences in genus 2. For example, we prove that the number of rational points P∈ C(ℚ) with h(P)≫g h(C) on C/ℚ smooth projective and of genus g≥ 2 is ≪ 1.872rank(Jac(C)(ℚ)), and that in fact the base of the exponent can be reduced to 1.311 once g≫ 1, though this is surely known to experts (the difference is the use of the Kabatiansky-Levenshtein bound in lieu of more elementary techniques).