TY - JOUR AB - We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split quadric surface AU - Browning, Timothy D AU - Lyczak, Julian AU - Sarapin, Roman ID - 13180 IS - 2 JF - Involve SN - 1944-4176 TI - Local solubility for a family of quadrics over a split quadric surface VL - 16 ER - TY - JOUR AB - We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle. AU - Browning, Timothy D AU - Sawin, Will ID - 13091 IS - 3 JF - Algebra and Number Theory SN - 1937-0652 TI - Free rational curves on low degree hypersurfaces and the circle method VL - 17 ER - TY - JOUR 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 IS - 3 JF - Annals of Mathematics SN - 0003-486X TI - The Hasse principle for random Fano hypersurfaces VL - 197 ER - TY - JOUR AB - We apply a variant of the square-sieve to produce an upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over P1 whose general fibre is a hyperelliptic curve. The implied constant does not depend on the coefficients of the polynomial defining the surface. AU - Bonolis, Dante AU - Browning, Timothy D ID - 12916 IS - 1 JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze SN - 0391-173X TI - Uniform bounds for rational points on hyperelliptic fibrations VL - 24 ER - TY - JOUR 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 IS - 10 JF - Algebra & Number Theory SN - 1937-0652 TI - Equidistribution and freeness on Grassmannians VL - 16 ER - TY - JOUR AB - An improved asymptotic formula is established for the number of rational points of bounded height on the split smooth del Pezzo surface of degree 5. The proof uses the five conic bundle structures on the surface. AU - Browning, Timothy D ID - 12776 JF - New York Journal of Mathematics SN - 1076-9803 TI - Revisiting the Manin–Peyre conjecture for the split del Pezzo surface of degree 5 VL - 28 ER - TY - BOOK AB - The Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties. This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. AU - Browning, Timothy D ID - 10415 SN - 0743-1643 TI - Cubic Forms and the Circle Method VL - 343 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 - 0025-5874 TI - Arithmetic of higher-dimensional orbifolds and a mixed Waring problem VL - 299 ER - 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 - 0933-7741 TI - The geometric sieve for quadrics VL - 33 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 - 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 - 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 SN - 0012-7094 TI - Density of rational points on a quadric bundle in ℙ3×ℙ3 VL - 169 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 show that a twisted variant of Linnik’s conjecture on sums of Kloosterman sums leads to an optimal covering exponent for S3. AU - Browning, Timothy D AU - Kumaraswamy, Vinay AU - Steiner, Rapael ID - 169 JF - International Mathematics Research Notices TI - Twisted Linnik implies optimal covering exponent for S3 ER - TY - JOUR AB - 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. AU - Browning, Timothy D AU - Schindler, Damaris ID - 172 JF - International Mathematics Research Notices TI - Strong approximation and a conjecture of Harpaz and Wittenberg ER -