@article{21002, abstract = {The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.}, author = {Browning, Timothy D}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{The Davenport–Heilbronn method: 80 years on}}, doi = {10.1112/jlms.70371}, volume = {113}, year = {2026}, } @article{21003, abstract = {We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2397-3129}, journal = {Discrete Analysis}, publisher = {Cambridge: Alliance of Diamond Open Access Journals}, title = {{Counting integer points on affine surfaces with a side condition}}, doi = {10.19086/da.143787}, volume = {2025}, year = {2025}, } @article{21266, abstract = {For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {European Mathematical Society Press}, title = {{Almost all quadratic twists of an elliptic curve have no integral points}}, doi = {10.4171/jems/1704}, year = {2025}, } @article{21343, abstract = {The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x], such that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q. Given a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all quadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common root in Fp. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {2270-518X}, journal = {Journal de l'ecole polytechnique mathematiques}, pages = {1677--1691}, publisher = {Ecole polytechnique}, title = {{Solubility of a resultant equation and applications}}, doi = {10.5802/jep.320}, volume = {12}, year = {2025}, } @article{21768, abstract = {Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes p. We give a different formulation of this result. More precisely, we replace dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch if HF3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.}, author = {Browning, Timothy D and Glas, Jakob and Wang, Victor}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {4}, publisher = {Springer Nature}, title = {{Optimal sums of three cubes in Fq[t]}}, doi = {10.1007/s00209-025-03765-z}, volume = {310}, year = {2025}, } @article{20249, abstract = {We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.}, author = {Browning, Timothy D and Wilsch, Florian Alexander}, issn = {1420-9020}, journal = {Selecta Mathematica New Series}, number = {4}, publisher = {Springer Nature}, title = {{Integral points on cubic surfaces: heuristics and numerics}}, doi = {10.1007/s00029-025-01074-1}, volume = {31}, year = {2025}, } @article{20367, abstract = {We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.}, author = {Browning, Timothy D and Sawin, Will and Wang, Victor}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {1863–1880}, publisher = {Springer Nature}, title = {{Pairs of commuting integer matrices}}, doi = {10.1007/s00208-025-03285-5}, volume = {393}, year = {2025}, } @article{17323, abstract = {We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2041-7942}, journal = {Mathematika}, number = {4}, publisher = {London Mathematical Society}, title = {{Strong divisibility sequences and sieve methods}}, doi = {10.1112/mtk.12269}, volume = {70}, year = {2024}, } @phdthesis{18132, abstract = {In this thesis, we are dealing with both arithmetic and geometric problems coming from the study of rational points with a particular focus on function fields over finite fields: (1) Using the circle method we produce upper bounds for the number of rational points of bounded height on diagonal cubic surfaces and fourfolds over Fq(t). This is based on joint work with Leonhard Hochfilzer. (2) We study rational points on smooth complete intersections X defined by cubic and quadratic hypersurfaces over Fq(t). We refine the Farey dissection of the “unit square” developed by Vishe [202] and use the circle method with a Kloosterman refinement to establish an asymptotic formula for the number of rational points of bounded height on X when dim(X) ≥ 23. Under the same hypotheses, we also verify weak approximation. (3) In joint work with Hochfilzer, we obtain upper bounds for the number of rational points of bounded height on del Pezzo surfaces of low degree over any global field. Our approach is to take hyperplane sections, which reduces the problem to uniform estimates for the number of rational points on curves. (4) We develop a version of the circle method capable of counting Fq-points on jet schemes of moduli spaces of rational curves on hypersurfaces. Combining this with a spreading out argument and a result of Mustaţă [150], this allows us to show that these moduli spaces only have canonical singularities under suitable assumptions on the degree and the dimension. In addition, we give an overview of guiding questions and conjectures in the field of rational points and explain the basic mechanism underlying the circle method. }, author = {Glas, Jakob}, issn = {2663-337X}, pages = {195}, publisher = {Institute of Science and Technology Austria}, title = {{Counting rational points over function fields}}, doi = {10.15479/at:ista:18132}, year = {2024}, } @article{18173, abstract = {Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided char (Fq)>3. Under the same hypotheses, we also verify weak approximation.}, author = {Glas, Jakob}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {London Mathematical Society}, title = {{Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t)}}, doi = {10.1112/jlms.12991}, volume = {110}, year = {2024}, } @unpublished{18295, abstract = {By developing a suitable version of the circle method, we show that the space of degree e rational curves on a smooth hypersurface of degree d has only canonical singularities provided its dimension is sufficiently large with respect to e and d.}, author = {Glas, Jakob}, booktitle = {arXiv}, title = {{Canonical singularities on moduli spaces of rational curves via the circle method}}, doi = {10.48550/arXiv.2405.16648}, year = {2024}, }