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2020 The dimension growth conjecture, polynomial in the degree and without logarithmic factors
Wouter Castryck, Raf Cluckers, Philip Dittmann, Kien Huu Nguyen
Algebra Number Theory 14(8): 2261-2294 (2020). DOI: 10.2140/ant.2020.14.2261

Abstract

We study Heath-Brown’s and Serre’s dimension growth conjecture (proved by Salberger) when the degree d grows. Recall that Salberger’s dimension growth results give bounds of the form O X , 𝜀 ( B dim X + 𝜀 ) for the number of rational points of height at most B on any integral subvariety X of n of degree d 2 , where one can write O d , n , 𝜀 instead of O X , 𝜀 as soon as d 4 . We give the following simplified and strengthened forms of these results: we remove the factor B 𝜀 as soon as d 5 , we obtain polynomial dependence on d of the implied constant, and we give a simplified, self-contained approach for d 1 6 . Along the way, we improve the well-known bounds due to Bombieri and Pila on the number of integral points of bounded height on affine curves and those by Walsh on the number of rational points of bounded height on projective curves. This leads to a slight sharpening of a recent estimate due to Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the 2 -torsion subgroup of the class group of a degree d number field. Our treatment builds on recent work by Salberger, who brings in many primes in Heath-Brown’s variant of the determinant method, and on recent work by Walsh and by Ellenberg and Venkatesh who bring in the size of the defining polynomial. We also obtain lower bounds showing that one cannot do better than polynomial dependence on d .

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Wouter Castryck. Raf Cluckers. Philip Dittmann. Kien Huu Nguyen. "The dimension growth conjecture, polynomial in the degree and without logarithmic factors." Algebra Number Theory 14 (8) 2261 - 2294, 2020. https://doi.org/10.2140/ant.2020.14.2261

Information

Received: 4 February 2020; Accepted: 23 April 2020; Published: 2020
First available in Project Euclid: 12 November 2020

MathSciNet: MR4172708
Digital Object Identifier: 10.2140/ant.2020.14.2261

Subjects:
Primary: 11D45
Secondary: 11G35 , 14G05

Keywords: dimension growth conjecture , rational points of bounded height

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.14 • No. 8 • 2020
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