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). AU - Bonolis, Dante ID - 9364 IS - 3 JF - Mathematical Proceedings of the Cambridge Philosophical Society SN - 0305-0041 TI - On the size of the maximum of incomplete Kloosterman sums VL - 172 ER - TY - JOUR AB - 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. AU - Cao, Yang AU - Huang, Zhizhong ID - 10765 IS - 3 JF - Advances in Mathematics SN - 0001-8708 TI - Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics VL - 398 ER - TY - GEN AB - We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant $\alpha$ and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties. AU - Wilsch, Florian Alexander ID - 10788 KW - Integral point KW - toric variety KW - Manin's conjecture T2 - arXiv TI - Integral points of bounded height on a certain toric variety ER - TY - THES AB - In this thesis, we study two of the most important questions in Arithmetic geometry: that of the existence and density of solutions to Diophantine equations. In order for a Diophantine equation to have any solutions over the rational numbers, it must have solutions everywhere locally, i.e., over R and over Qp for every prime p. The converse, called the Hasse principle, is known to fail in general. However, it is still a central question in Arithmetic geometry to determine for which varieties the Hasse principle does hold. In this work, we establish the Hasse principle for a wide new family of varieties of the form f(t) = NK/Q(x) ̸= 0, where f is a polynomial with integer coefficients and NK/Q denotes the norm form associated to a number field K. Our results cover products of arbitrarily many linear, quadratic or cubic factors, and generalise an argument of Irving [69], which makes use of the beta sieve of Rosser and Iwaniec. We also demonstrate how our main sieve results can be applied to treat new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations. In the second question, about the density of solutions, one defines a height function and seeks to estimate asymptotically the number of points of height bounded by B as B → ∞. Traditionally, one either counts rational points, or integral points with respect to a suitable model. However, in this thesis, we study an emerging area of interest in Arithmetic geometry known as Campana points, which in some sense interpolate between rational and integral points. More precisely, we count the number of nonzero integers z1, z2, z3 such that gcd(z1, z2, z3) = 1, and z1, z2, z3, z1 + z2 + z3 are all squareful and bounded by B. Using the circle method, we obtain an asymptotic formula which agrees in the power of B and log B with a bold new generalisation of Manin’s conjecture to the setting of Campana points, recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [96]. However, in this thesis we also provide the first known counterexamples to leading constant predicted by their conjecture. AU - Shute, Alec L ID - 12072 SN - 2663-337X TI - Existence and density problems in Diophantine geometry: From norm forms to Campana points ER - TY - JOUR AB - 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. AU - Kmentt, Philip AU - Shute, Alec L ID - 11636 IS - 10 JF - Finite Fields and their Applications SN - 10715797 TI - The Bertini irreducibility theorem for higher codimensional slices VL - 83 ER - TY - JOUR AB - 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. AU - Derenthal, Ulrich AU - Wilsch, Florian Alexander ID - 10018 JF - Journal of the Institute of Mathematics of Jussieu KW - Integral points KW - del Pezzo surface KW - universal torsor KW - Manin’s conjecture SN - 1474-7480 TI - Integral points on singular del Pezzo surfaces ER - TY - JOUR AB - 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. AU - Autissier, Pascal AU - Bonolis, Dante AU - Lamzouri, Youness ID - 10711 IS - 7 JF - Compositio Mathematica KW - Algebra and Number Theory SN - 0010-437X TI - The distribution of the maximum of partial sums of Kloosterman sums and other trace functions VL - 157 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 - 09337741 TI - The geometric sieve for quadrics VL - 33 ER - TY - GEN AB - 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. AU - Wilsch, Florian Alexander ID - 9034 T2 - arXiv TI - Integral points of bounded height on a log Fano threefold 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 - 0025-5874 TI - Arithmetic of higher-dimensional orbifolds and a mixed Waring problem VL - 299 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 - GEN AB - We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold $(\mathbb{P}^1,D)$, where $D =\frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty]$. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms. AU - Shute, Alec L ID - 12077 T2 - arXiv TI - On the leading constant in the Manin-type conjecture for Campana points ER - TY - GEN AB - We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado. AU - Shute, Alec L ID - 12076 T2 - arXiv TI - Sums of four squareful numbers 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 - 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 - 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 - 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. AU - Ionica, Sorina AU - Kılıçer, Pınar AU - Lauter, Kristin AU - Lorenzo García, Elisa AU - Manzateanu, Maria-Adelina AU - Massierer, Maike AU - Vincent, Christelle ID - 10874 JF - Research in Number Theory KW - Algebra and Number Theory SN - 2522-0160 TI - Modular invariants for genus 3 hyperelliptic curves VL - 5 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 - JOUR AB - 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. AU - De La Bretèche, Régis AU - Destagnol, Kevin N AU - Liu, Jianya AU - Wu, Jie AU - Zhao, Yongqiang ID - 6620 IS - 12 JF - Science China Mathematics SN - 16747283 TI - On a certain non-split cubic surface VL - 62 ER - TY - JOUR AB - 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. AU - Destagnol, Kevin N AU - Sofos, Efthymios ID - 6835 IS - 11 JF - Bulletin des Sciences Mathematiques SN - 0007-4497 TI - Rational points and prime values of polynomials in moderately many variables VL - 156 ER -