0 there exists a large subset of a ∈ F×p such that for kl a,1,p : x → e((ax+x) / p) we have M(kla,1,p) ≥ (1−ε/√2π + o(1)) log log p, as p→∞. Finally, we prove a result on the growth of the moments of {M (kla,1,p)}a∈F×p. 2020 Mathematics Subject Classification: 11L03, 11T23 (Primary); 14F20, 60F10 (Secondary).}, author = {Bonolis, Dante}, issn = {1469-8064}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, number = {3}, pages = {563 -- 590}, publisher = {Cambridge University Press}, title = {{On the size of the maximum of incomplete Kloosterman sums}}, doi = {10.1017/S030500412100030X}, volume = {172}, year = {2022}, } @article{11636, abstract = {In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.}, author = {Kmentt, Philip and Shute, Alec L}, issn = {10902465}, journal = {Finite Fields and their Applications}, number = {10}, publisher = {Elsevier}, title = {{The Bertini irreducibility theorem for higher codimensional slices}}, doi = {10.1016/j.ffa.2022.102085}, volume = {83}, year = {2022}, } @article{10765, abstract = {We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,...,xn)=m, where q is a non-degenerate integral quadratic form in n>3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.}, author = {Cao, Yang and Huang, Zhizhong}, issn = {1090-2082}, journal = {Advances in Mathematics}, number = {3}, publisher = {Elsevier}, title = {{Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics}}, doi = {10.1016/j.aim.2022.108236}, volume = {398}, year = {2022}, } @article{10711, abstract = {In this paper, we investigate the distribution of the maximum of partial sums of families of m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.}, author = {Autissier, Pascal and Bonolis, Dante and Lamzouri, Youness}, issn = {1570-5846}, journal = {Compositio Mathematica}, keywords = {Algebra and Number Theory}, number = {7}, pages = {1610--1651}, publisher = {Cambridge University Press}, title = {{The distribution of the maximum of partial sums of Kloosterman sums and other trace functions}}, doi = {10.1112/s0010437x21007351}, volume = {157}, year = {2021}, } @article{8742, abstract = {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.}, author = {Browning, Timothy D and Heath-Brown, Roger}, issn = {14355337}, journal = {Forum Mathematicum}, number = {1}, pages = {147--165}, publisher = {De Gruyter}, title = {{The geometric sieve for quadrics}}, doi = {10.1515/forum-2020-0074}, volume = {33}, year = {2021}, } @unpublished{9034, abstract = {We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of $\mathbb{P}^3$ outside certain planes using universal torsors.}, author = {Wilsch, Florian Alexander}, booktitle = {arXiv}, title = {{Integral points of bounded height on a log Fano threefold}}, year = {2021}, } @unpublished{9199, abstract = {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.}, author = {Browning, Timothy D and Horesh, Tal and Wilsch, Florian Alexander}, booktitle = {arXiv}, title = {{Equidistribution and freeness on Grassmannians}}, year = {2021}, } @article{9260, abstract = {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.}, author = {Browning, Timothy D and Yamagishi, Shuntaro}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, pages = {1071–1101}, publisher = {Springer Nature}, title = {{Arithmetic of higher-dimensional orbifolds and a mixed Waring problem}}, doi = {10.1007/s00209-021-02695-w}, volume = {299}, year = {2021}, } @unpublished{10018, abstract = {In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type A1 + A3 and prove an analogue of Manin's conjecture for integral points with respect to its singularities and its lines.}, author = {Derenthal, Ulrich and Wilsch, Florian Alexander}, booktitle = {arXiv}, keywords = {Integral points, del Pezzo surface, universal torsor, Manin’s conjecture}, title = {{Integral points on singular del Pezzo surfaces}}, year = {2021}, } @book{10415, abstract = {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.}, author = {Browning, Timothy D}, isbn = {978-3-030-86871-0}, issn = {2296-505X}, pages = {XIV, 166}, publisher = {Springer Nature}, title = {{Cubic Forms and the Circle Method}}, doi = {10.1007/978-3-030-86872-7}, volume = {343}, year = {2021}, } @article{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 = {Browning, Timothy D and Sawin, Will}, journal = {Annals of Mathematics}, number = {3}, pages = {893--948}, publisher = {Princeton University}, title = {{A geometric version of the circle method}}, doi = {10.4007/annals.2020.191.3.4}, volume = {191}, year = {2020}, } @article{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}, journal = {Duke Mathematical Journal}, number = {16}, pages = {3099--3165}, publisher = {Project Euclid}, title = {{Density of rational points on a quadric bundle in ℙ3×ℙ3}}, doi = {10.1215/00127094-2020-0031}, volume = {169}, year = {2020}, } @unpublished{8682, abstract = {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.}, author = {Browning, Timothy D and Boudec, Pierre Le and Sawin, Will}, booktitle = {arXiv}, title = {{The Hasse principle for random Fano hypersurfaces}}, year = {2020}, } @article{9007, abstract = {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.}, author = {Browning, Timothy D and Sawin, Will}, issn = {14208946}, journal = {Commentarii Mathematici Helvetici}, number = {4}, pages = {635--659}, publisher = {European Mathematical Society}, title = {{Free rational points on smooth hypersurfaces}}, doi = {10.4171/CMH/499}, volume = {95}, year = {2020}, } @article{10874, abstract = {In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.}, author = {Ionica, Sorina and Kılıçer, Pınar and Lauter, Kristin and Lorenzo García, Elisa and Manzateanu, Maria-Adelina and Massierer, Maike and Vincent, Christelle}, issn = {2522-0160}, journal = {Research in Number Theory}, keywords = {Algebra and Number Theory}, publisher = {Springer Nature}, title = {{Modular invariants for genus 3 hyperelliptic curves}}, doi = {10.1007/s40993-018-0146-6}, volume = {5}, 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}}, doi = {10.1090/tran/7514}, 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}, } @article{6620, abstract = {This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of ℙ3ℚ given by the following equation 𝑥0(𝑥21+𝑥22)−𝑥33=0 in agreement with the Manin-Peyre conjectures. }, author = {De La Bretèche, Régis and Destagnol, Kevin N and Liu, Jianya and Wu, Jie and Zhao, Yongqiang}, issn = {16747283}, journal = {Science China Mathematics}, number = {12}, pages = {2435–2446}, publisher = {Springer}, title = {{On a certain non-split cubic surface}}, doi = {10.1007/s11425-018-9543-8}, volume = {62}, year = {2019}, } @article{6835, abstract = {We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.}, author = {Destagnol, Kevin N and Sofos, Efthymios}, issn = {0007-4497}, journal = {Bulletin des Sciences Mathematiques}, number = {11}, publisher = {Elsevier}, title = {{Rational points and prime values of polynomials in moderately many variables}}, doi = {10.1016/j.bulsci.2019.102794}, volume = {156}, year = {2019}, }