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

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 $B^{\epsilon }$ factor by a $\log (B)$ factor.

v4: final version. Fixed some typos and errors pointed out by the referee. To appear in Algebra and Number Theory. 45 pages

References in corpus (3)

• Determinant method and the pseudo-effective threshold
• Points of bounded height on curves and the dimension growth conjecture over ${\mathbb{F}}_q[t]$
• On the global determinant method

Cited by in corpus (4)

• Improvements on dimension growth results and effective Hilbert's irreducibility theorem
• Counting rational points on smooth hypersurfaces with high degree
• Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth
• Points of bounded height on weighted projective spaces over global function fields