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Uniform bounds for the number of rational points on varieties over global fields (2101.12174v4)
Published 28 Jan 2021 in math.NT and math.AG
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 $B{\varepsilon}$ factor by a $\log(B)$ factor.