TY - JOUR
AB - We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.
AU - Browning, Timothy D
AU - Heath-Brown, Roger
ID - 8742
IS - 1
JF - Forum Mathematicum
SN - 09337741
TI - The geometric sieve for quadrics
VL - 33
ER -
TY - GEN
AB - We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano
varieties.
AU - Browning, Timothy D
AU - Horesh, Tal
AU - Wilsch, Florian Alexander
ID - 9199
T2 - arXiv
TI - Equidistribution and freeness on Grassmannians
ER -
TY - JOUR
AB - We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ) when Δ is a Q-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.
AU - Browning, Timothy D
AU - Yamagishi, Shuntaro
ID - 9260
JF - Mathematische Zeitschrift
SN - 00255874
TI - Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
ER -
TY - GEN
AB - It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least 3 over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least 3. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.
AU - Browning, Timothy D
AU - Boudec, Pierre Le
AU - Sawin, Will
ID - 8682
T2 - arXiv
TI - The Hasse principle for random Fano hypersurfaces
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Sawin, Will
ID - 177
IS - 3
JF - Annals of Mathematics
TI - A geometric version of the circle method
VL - 191
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Heath Brown, Roger
ID - 179
IS - 16
JF - Duke Mathematical Journal
TI - Density of rational points on a quadric bundle in ℙ3×ℙ3
VL - 169
ER -
TY - JOUR
AB - Motivated by a recent question of Peyre, we apply the Hardy–Littlewood circle method to count “sufficiently free” rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rationals.
AU - Browning, Timothy D
AU - Sawin, Will
ID - 9007
IS - 4
JF - Commentarii Mathematici Helvetici
SN - 00102571
TI - Free rational points on smooth hypersurfaces
VL - 95
ER -
TY - JOUR
AB - 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 .
AU - Browning, Timothy D
AU - Sofos, Efthymios
ID - 170
IS - 3-4
JF - Mathematische Annalen
TI - Counting rational points on quartic del Pezzo surfaces with a rational conic
VL - 373
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Loughran, Daniel
ID - 175
IS - 8
JF - Transactions of the American Mathematical Society
SN - 00029947
TI - Sieving rational points on varieties
VL - 371
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Hu, L.Q.
ID - 6310
JF - Advances in Mathematics
SN - 00018708
TI - Counting rational points on biquadratic hypersurfaces
VL - 349
ER -
TY - CONF
AB - We survey recent efforts to quantify failures of the Hasse principle in families of rationally connected varieties.
AU - Browning, Timothy D
ID - 174
IS - 2
TI - How often does the Hasse principle hold?
VL - 97
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Sofos, Efthymios
ID - 176
IS - 3
JF - International Journal of Nuber Theory
TI - Averages of arithmetic functions over principal ideals
VL - 15
ER -
TY - JOUR
AB - 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.
AU - Browning, Timothy D
AU - Heath-Brown, Roger
ID - 178
JF - Discrete Analysis
TI - Counting rational points on quadric surfaces
VL - 15
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
AU - Vishe, Pankaj
ID - 265
IS - 7
JF - Geometric Methods in Algebra and Number Theory
TI - Rational curves on smooth hypersurfaces of low degree
VL - 11
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
AU - Heath-Brown, Roger
ID - 266
IS - 2
JF - Journal of the European Mathematical Society
TI - Forms in many variables and differing degrees
VL - 19
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
ID - 267
IS - 3
JF - Mathematika
TI - Many cubic surfaces contain rational points
VL - 63
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
AU - Prendiville, Sean M
ID - 268
IS - 7
JF - International Mathematics Research Notices
TI - A transference approach to a Roth-type theorem in the squares
VL - 2017
ER -
TY - JOUR
AU - Browning, Timothy D
AU - Loughran, Daniel
ID - 269
IS - 3-4
JF - Mathematische Zeitschrift
TI - Varieties with too many rational points
VL - 285
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
AU - Gorodnik, Alexander
ID - 270
IS - 6
JF - Proceedings of the London Mathematical Society
TI - Power-free values of polynomials on symmetric varieties
VL - 114
ER -
TY - JOUR
AB - 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.
AU - Timothy Browning
AU - Prendiville, Sean M
ID - 271
IS - 731
JF - Journal fur die Reine und Angewandte Mathematik
TI - Improvements in Birch's theorem on forms in many variables
VL - 2017
ER -