Uniform bounds for the number of rational points on varieties over global fields

Paredes, Marcelo; Sasyk, Román

doi:10.2140/ant.2022.16.1941

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Uniform bounds for the number of rational points on varieties over global fields

Marcelo Paredes and Román Sasyk

Vol. 16 (2022), No. 8, 1941–2000

DOI: 10.2140/ant.2022.16.1941

Abstract

We extend the work of Salberger; Walsh; Castryck, Cluckers, Dittmann and Nguyen; and Vermeulen to prove the uniform dimension growth conjecture of Heath-Brown and Serre for varieties of degree at least $4$ over global fields. As an intermediate step, we generalize the bounds of Bombieri and Pila to curves over global fields and in doing so we improve the $B^{𝜀}$ factor by a $\log (B)$ factor.

Keywords

varieties over global fields, heights in global fields, number of rational solutions of diophantine equations, determinant method

Mathematical Subject Classification

Primary: 11D45, 11G35, 11G50, 14G05

Milestones

Received: 28 February 2021

Revised: 14 August 2021

Accepted: 12 November 2021

Published: 29 November 2022

Authors

Marcelo Paredes
Department of Mathematics Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria – Pabellón I 1428 Ciudad Autónoma de Buenos Aires Argentina

Román Sasyk
Department of Mathematics Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria – Pabellón I 1428 Ciudad Autónoma de Buenos Aires Argentina	Instituto Argentino de Matemáticas Alberto P. Calderón-CONICET Saavedra 15, Piso 3 1083 Ciudad Autónoma de Buenos Aires Argentina

Vol. 16, No. 8, 2022