# Uniform bounds for rational points on hyperelliptic fibrations

@article{Bonolis2021UniformBF,
title={Uniform bounds for rational points on hyperelliptic fibrations},
author={Dante Bonolis and Tim D. Browning},
journal={ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE},
year={2021}
}
• Published 28 July 2020
• Mathematics
• ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic curve.
1 Citations
Diagonal cubic forms and the large sieve
Let F (x) be a diagonal integer-coefficient cubic form in m ∈ {4, 5, 6} variables. Excluding rational lines if m = 4, we bound the number of integral solutions x ∈ [−X,X] to F (x) = 0 by OF, (X

#### References

SHOWING 1-10 OF 24 REFERENCES
Counting rational points on smooth cyclic covers
• Mathematics
• 2011
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn−1. In this paper, we achieve Serreʼs conjecture in the special case of smooth
On the number of points on a complete intersection over a finite field
Abstract Let V be a projective complete intersection defined over the finite field F q of q = p α elements and suppose it has dimension n and a singular locus of dimension d . We prove that the
Integral points on elliptic curves and 3-torsion in class groups
• Mathematics
• 2004
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on
Rational points of bounded height on Fano varieties
• Mathematics
• 1989
a prime pe7Z. Let V be an algebraic variety defined over F and lI~ a metrized line bundle on V, i.e., a system (L, ]'],) consisting of a line bundle L and a family of Banach v-adic metrics on L | F,,
A bound for the 3-part of class numbers of quadratic fields by means of the square sieve
Abstract We prove a nontrivial bound of O(|D|27/56+ε) for the 3-part of the class number of a quadratic field ℚ(√D) by using a variant of the square sieve and the q-analogue of van der Corput's
The polynomial sieve and equal sums of like polynomials
A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.
The number of integral points on arcs and ovals
• Mathematics
• 1989
integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed,
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• Mathematics
• 2017
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being \$O_{\epsilon}(|{\rm
Integral Points on Elliptic Curves and the Bombieri-Pila Bounds
Let C be an affine, plane, algebraic curve of degree d with integer coefficients. In 1989, Bombieri and Pila showed that if one takes a box with sides of length N then C can obtain no more than
On the Rationality of the Zeta Function of an Algebraic Variety
Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its