@article{170,
abstract = {Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over ℚ that contains a conic defined over ℚ .},
author = {Browning, Timothy D and Sofos, Efthymios},
journal = {Mathematische Annalen},
number = {3-4},
pages = {977--1016},
publisher = {Springer Nature},
title = {{Counting rational points on quartic del Pezzo surfaces with a rational conic}},
doi = {10.1007/s00208-018-1716-6},
volume = {373},
year = {2019},
}
@article{175,
abstract = {An upper bound sieve for rational points on suitable varieties isdeveloped, together with applications tocounting rational points in thin sets,to local solubility in families, and to the notion of “friable” rational pointswith respect to divisors. In the special case of quadrics, sharper estimates areobtained by developing a version of the Selberg sieve for rational points.},
author = {Browning, Timothy D and Loughran, Daniel},
issn = {10886850},
journal = {Transactions of the American Mathematical Society},
number = {8},
pages = {5757--5785},
publisher = {American Mathematical Society},
title = {{Sieving rational points on varieties}},
volume = {371},
year = {2019},
}
@article{6310,
abstract = {An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.},
author = {Browning, Timothy D and Hu, L.Q.},
issn = {10902082},
journal = {Advances in Mathematics},
pages = {920--940},
publisher = {Elsevier},
title = {{ Counting rational points on biquadratic hypersurfaces}},
doi = {10.1016/j.aim.2019.04.031},
volume = {349},
year = {2019},
}
@inproceedings{174,
abstract = {We survey recent efforts to quantify failures of the Hasse principle in families of rationally connected varieties.},
author = {Browning, Timothy D},
location = {Salt Lake City, Utah, USA},
number = {2},
pages = {89 -- 102},
publisher = {American Mathematical Society},
title = {{How often does the Hasse principle hold?}},
doi = {10.1090/pspum/097.2/01700},
volume = {97},
year = {2018},
}
@unpublished{179,
abstract = {An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface x1y21+⋯+x4y24=0 in ℙ3×ℙ3. This confirms the modified Manin conjecture for this variety, in which the removal of a thin set of rational points is allowed.},
author = {Browning, Timothy D and Heath Brown, Roger},
booktitle = {Unknown},
pages = {1 -- 60},
publisher = {Unknown},
title = {{Density of rational points on a quadric bundle in ℙ3×ℙ3}},
year = {2018},
}
@unpublished{176,
abstract = {For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with integer coefficients.},
author = {Browning, Timothy D and Sofos, Efthymios},
booktitle = {International Journal of Nuber Theory},
pages = {1 -- 22},
publisher = {World Scientific Publishing},
title = {{Averages of arithmetic functions over principal ideals}},
year = {2018},
}
@article{178,
abstract = {We give an upper bound for the number of rational points of height at most B, lying on a surface defined by a quadratic form Q. The bound shows an explicit dependence on Q. It is optimal with respect to B, and is also optimal for typical forms Q.},
author = {Browning, Timothy D and Heath-Brown, Roger},
issn = {2397-3129},
journal = {Discrete Analysis},
pages = {1 -- 31},
publisher = {Alliance of Diamond Open Access Journals},
title = {{Counting rational points on quadric surfaces}},
doi = {10.19086/da.4375},
year = {2018},
}
@article{268,
abstract = {We show that any subset of the squares of positive relative upper density contains nontrivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.},
author = {Timothy Browning and Prendiville, Sean M},
journal = {International Mathematics Research Notices},
number = {7},
pages = {2219 -- 2248},
publisher = {Oxford University Press},
title = {{A transference approach to a Roth-type theorem in the squares}},
doi = {10.1093/imrn/rnw096},
volume = {2017},
year = {2017},
}
@article{270,
abstract = {Given a symmetric variety Y defined over Q and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y . We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over Q.},
author = {Timothy Browning and Gorodnik, Alexander},
journal = {Proceedings of the London Mathematical Society},
number = {6},
pages = {1044 -- 1080},
publisher = {Wiley Blackwell},
title = {{Power-free values of polynomials on symmetric varieties}},
doi = {10.1112/plms.12030},
volume = {114},
year = {2017},
}
@article{269,
author = {Browning, Timothy D and Loughran, Daniel},
journal = {Mathematische Zeitschrift},
number = {3-4},
pages = {1249 -- 1267},
publisher = {Springer},
title = {{Varieties with too many rational points}},
doi = {10.1007/s00209-016-1746-2},
volume = {285},
year = {2017},
}
@article{271,
abstract = {We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the form has at least (d-\sqrt{d}/2)2^d variables. This improves on a longstanding result of Birch.},
author = {Timothy Browning and Prendiville, Sean M},
journal = {Journal fur die Reine und Angewandte Mathematik},
number = {731},
pages = {203 -- 234},
publisher = {Walter de Gruyter},
title = {{Improvements in Birch's theorem on forms in many variables}},
doi = {doi.org/10.1515/crelle-2014-0122},
volume = {2017},
year = {2017},
}
@article{169,
abstract = {We show that a twisted variant of Linnik’s conjecture on sums of Kloosterman sums leads to an optimal covering exponent for S3.},
author = {Timothy Browning and Kumaraswamy, Vinay V and Steiner, Rapael S},
journal = {International Mathematics Research Notices},
publisher = {Oxford University Press},
title = {{Twisted Linnik implies optimal covering exponent for S3}},
doi = {https://doi.org/10.1093/imrn/rnx116},
year = {2017},
}
@article{265,
abstract = {We establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.},
author = {Timothy Browning and Vishe, Pankaj},
journal = {Geometric Methods in Algebra and Number Theory},
number = {7},
pages = {1657 -- 1675},
publisher = { Mathematical Sciences Publishers},
title = {{Rational curves on smooth hypersurfaces of low degree}},
doi = {10.2140/ant.2017.11.1657},
volume = {11},
year = {2017},
}
@article{272,
abstract = {Given a number field K/Q and a polynomial P ε Q [t], all of whose roots are Q, let X be the variety defined by the equation NK (x) = P (t). Combining additive combinatiorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hesse principle and weak approximation on any smooth and projective model of X.},
author = {Timothy Browning and Matthiesen, Lilian},
journal = {Annales Scientifiques de l'Ecole Normale Superieure},
number = {6},
pages = {1383 -- 1446},
publisher = {Societe Mathematique de France},
title = {{Norm forms for arbitrary number fields as products of linear polynomials}},
doi = {10.24033/asens.2348},
volume = {50},
year = {2017},
}
@unpublished{177,
abstract = {We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.},
author = {Timothy Browning and Sawin, Will},
booktitle = {Unknown},
pages = {1 -- 47},
publisher = {Unknown},
title = {{A geometric version of the circle method}},
year = {2017},
}
@article{172,
abstract = {We study strong approximation for some algebraic varieties over ℚ which are defined using norm forms. This allows us to confirm a special case of a conjecture due to Harpaz and Wittenberg.},
author = {Timothy Browning and Schindler, Damaris},
journal = {International Mathematics Research Notices},
publisher = {Oxford University Press},
title = {{Strong approximation and a conjecture of Harpaz and Wittenberg}},
doi = {https://doi.org/10.1093/imrn/rnx252},
year = {2017},
}
@article{266,
abstract = {We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral variety X Pm, provided only that its dimension is large enough in terms of its degree.},
author = {Timothy Browning and Heath-Brown, Roger},
journal = {Journal of the European Mathematical Society},
number = {2},
pages = {357 -- 394},
publisher = {European Mathematical Society Publishing House},
title = {{Forms in many variables and differing degrees}},
doi = {10.4171/JEMS/668},
volume = {19},
year = {2017},
}
@article{267,
abstract = {Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.},
author = {Timothy Browning},
journal = {Mathematika},
number = {3},
pages = {818 -- 839},
publisher = {Cambridge University Press},
title = {{Many cubic surfaces contain rational points}},
doi = {10.1112/S0025579317000195},
volume = {63},
year = {2017},
}
@article{263,
abstract = {We count rational points of bounded height on the Cayley ruled cubic surface and interpret the result in the context of general conjectures due to Batyrev and Tschinkel.},
author = {de la Bretèche, Régis and Timothy Browning and Salberger, Per},
journal = {European Journal of Mathematics},
number = {1},
pages = {55 -- 72},
publisher = {Springer Nature},
title = {{Counting rational points on the Cayley ruled cubic}},
doi = {10.1007/s40879-015-0049-1},
volume = {2},
year = {2016},
}
@article{264,
abstract = {Given a family of varieties over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.},
author = {Bright, Maritn J and Timothy Browning and Loughran, Daniel},
journal = {Compositio Mathematica},
number = {7},
pages = {1435 -- 1475},
publisher = {Cambridge University Press},
title = {{Failures of weak approximation in families}},
doi = {10.1112/S0010437X16007405},
volume = {152},
year = {2016},
}