MANY CUBIC SURFACES CONTAIN RATIONAL POINTS

2017/01/01 by T. D. Browning
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Cubic form #Cubic function #Cubic surface #Field (mathematics) #Geometry #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Physics #Pure mathematics #Rational point #Thermodynamics #Work (physics) #math.NT #msc:11G35 #msc:14G05

paper · pdf · doi:10.1112/s0025579317000195

published as Mathematika 63 (2017) 818-839 · 23 pages; minor edits and added new remark (Remark 2.1) following an argument of Jahnel

openalex publication_date 2017/01/01 · arxiv created 2017/06/12 · arxiv updated 2017/12/06 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05

Abstract

Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

Citations

• Heegner points at Eisenstein primes and twists of elliptic curves
• Heegner points and derivatives ofL-series
• Failures of weak approximation in families
• The Arithmetic of Elliptic Curves