@article{19727, abstract = {By studying some Clausen-like multiple Dirichlet series, we complete the proof of Manin's conjecture for sufficiently split smooth equivariant compactifications of the translation-dilation group over the rationals. Secondary terms remain elusive in general.}, author = {Wang, Victor}, issn = {1090-2082}, journal = {Advances in Mathematics}, publisher = {Elsevier}, title = {{Asymptotic growth of translation-dilation orbits}}, doi = {10.1016/j.aim.2025.110341}, volume = {475}, year = {2025}, } @article{19673, abstract = {We show that almost all primes p =\= ± 4 mod9 are sums of three cubes, assuming a conjecture due to Hooley, Manin, et al. on cubic fourfolds. This conjecture is approachable under standard statistical hypotheses on geometric families of L-functions.}, author = {Wang, Victor}, issn = {2998-4114}, journal = {Journal of the Association for Mathematical Research}, number = {1}, pages = {1--26}, publisher = {Association for Mathematical Research}, title = {{Prime Hasse principles via diophantine second moments}}, doi = {10.56994/JAMR.003.001.001}, volume = {3}, year = {2025}, } @article{19407, abstract = {We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over Fq . Among other results, this allows us to prove that the Q-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.}, author = {Ballini, Francesco and Lombardo, Davide and Verzobio, Matteo}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, publisher = {Cambridge University Press}, title = {{On the L-polynomials of curves over finite fields}}, doi = {10.1017/prm.2025.7}, year = {2025}, } @article{19483, abstract = {We prove matching upper and lower bounds for the average of the6-torsionof class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.}, author = {Chan, Yik Tung and Koymans, Peter and Pagano, Carlo and Sofos, Efthymios}, issn = {2036-2145}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, publisher = {Scuola Normale Superiore - Edizioni della Normale}, title = {{6-torision and integral points on quartic threefolds}}, doi = {10.2422/2036-2145.202412_006}, year = {2025}, } @article{20222, abstract = {Let X be a smooth projective hypersurface defined over Q. We provide new bounds for rational points of bounded height on X. In particular, we show that if X is a smooth projective hypersurface in Pn with n 4 and degree d 50, then the set of rational points on X of height bounded by B have cardinality On,d,ε (Bn−2+ε ). If X is smooth and has degree d 6, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.}, author = {Verzobio, Matteo}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {16}, publisher = {Oxford University Press}, title = {{Counting rational points on smooth hypersurfaces with high degree}}, doi = {10.1093/imrn/rnaf249}, volume = {2025}, year = {2025}, } @article{20367, abstract = {We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.}, author = {Browning, Timothy D and Sawin, Will and Wang, Victor}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {1863–1880}, publisher = {Springer Nature}, title = {{Pairs of commuting integer matrices}}, doi = {10.1007/s00208-025-03285-5}, volume = {393}, year = {2025}, } @article{20423, abstract = {For any d 2, we prove that there exists an integer n0(d) such that there exists an n × n magic square of dth powers for all n n0(d). In particular, we establish the existence of an n × n magic square of squares for all n 4, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of n × n magic squares of dth powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.}, author = {Rome, Nick and Yamagishi, Shuntaro}, issn = {2363-9555}, journal = {Research in Number Theory}, number = {4}, publisher = {Springer Nature}, title = {{On the existence of magic squares of powers}}, doi = {10.1007/s40993-025-00671-5}, volume = {11}, year = {2025}, } @article{20603, abstract = {We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic progression of length 2k+1, and where G is a commutative group, in which every nonzero element has an order of at least 2k+1. More specifically, we show the following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression of length 2k+1, then |A+B|≥|A|2k−13k−2|B|k3k−2. As an application we derive upper bounds on the cardinality of the summands in sumsets A+B+C contained in the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5 for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3.}, author = {Elsholtz, Christian and Ruzsa, Imre Z. and Wurzinger, Lena}, issn = {1730-6264}, journal = {Acta Arithmetica}, pages = {289--303}, publisher = {Institute of Mathematics}, title = {{Sumset growth in progression-free sets}}, doi = {10.4064/aa250115-14-7}, volume = {220}, year = {2025}, } @article{18705, abstract = {Given a non-singular diagonal cubic hypersurface X⊂Pn−1 over Fq(t) with char(Fq)≠3, we show that the number of rational points of height at most |P| is O(|P|3+ε) for n=6 and O(|P|2+ε) for n=4. In fact, if n=4 and char(Fq)>3 we prove that the number of rational points away from any rational line contained in X is bounded by O(|P|3/2+ε). From the result in 6 variables we deduce weak approximation for diagonal cubic hypersurfaces for n≥7 over Fq(t) when char(Fq)>3 and handle Waring's problem for cubes in 7 variables over Fq(t) when char(Fq)≠3. Our results answer a question of Davenport regarding the number of solutions of bounded height to x31+x32+x33=x34+x35+x36 with xi∈Fq[t].}, author = {Glas, Jakob and Hochfilzer, Leonhard}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {5485--5533}, publisher = {Springer Nature}, title = {{On a question of Davenport and diagonal cubic forms over Fq(t)}}, doi = {10.1007/s00208-024-03035-z}, volume = {391}, year = {2025}, } @article{21343, abstract = {The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x], such that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q. Given a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all quadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common root in Fp. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {2270-518X}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {1677--1691}, publisher = {Ecole Polytechnique}, title = {{Solubility of a resultant equation and applications}}, doi = {10.5802/jep.320}, volume = {12}, year = {2025}, } @article{20249, abstract = {We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.}, author = {Browning, Timothy D and Wilsch, Florian Alexander}, issn = {1420-9020}, journal = {Selecta Mathematica New Series}, number = {4}, publisher = {Springer Nature}, title = {{Integral points on cubic surfaces: heuristics and numerics}}, doi = {10.1007/s00029-025-01074-1}, volume = {31}, year = {2025}, } @article{22929, abstract = {The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.}, author = {Bilu, Margaret and Browning, Timothy D}, issn = {1873-2151}, journal = {Annales Scientifiques de l’École Normale Supérieure}, keywords = {Circle method, moduli spaces of curves, hypersurfaces, Grothendieck ring of varieties, motivic integration}, number = {5}, pages = {1179--1242}, publisher = {Société Mathématique de France}, title = {{A motivic circle method}}, doi = {10.24033/asens.2628}, volume = {58}, year = {2025}, } @article{12312, abstract = {Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.}, author = {Lombardo, Davide and Verzobio, Matteo}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{On the local-global principle for isogenies of abelian surfaces}}, doi = {10.1007/s00029-023-00908-0}, volume = {30}, year = {2024}, } @article{17127, abstract = {Let P(x)∈Z[x] be a polynomial with at least two distinct complex roots. We prove that the number of solutions (x1,…,xk,y1,…,yk)∈[N]2k to the equation ∏1≤i≤kP(xi)=∏1≤j≤kP(yj)≠0 (for any k≥1 ) is asymptotically k!Nk as N→+∞. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums 1N√∑n≤Nf(P(n)) match standard complex Gaussian moments as N→+∞ , where f is the Steinhaus random multiplicative function.}, author = {Wang, Victor and Xu, Max Wenqiang}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {8}, pages = {2718--2726}, publisher = {London Mathematical Society}, title = {{Paucity phenomena for polynomial products}}, doi = {10.1112/blms.13095}, volume = {56}, year = {2024}, } @article{15338, abstract = {We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.}, author = {Browning, Timothy D and Pierce, Lillian B. and Schindler, Damaris}, issn = {1475-3030}, journal = {Journal of the Institute of Mathematics of Jussieu}, number = {6}, pages = {2859--2912}, publisher = {Cambridge University Press}, title = {{Generalised quadratic forms over totally real number fields}}, doi = {10.1017/S1474748024000161}, volume = {23}, year = {2024}, } @article{15337, abstract = {We prove the Manin–Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1, 2).}, author = {Bonolis, Dante and Browning, Timothy D and Huang, Zhizhong}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {4123--4207}, publisher = {Springer Nature}, title = {{Density of rational points on some quadric bundle threefolds}}, doi = {10.1007/s00208-024-02854-4}, volume = {390}, year = {2024}, } @article{18930, abstract = {We study sumsets 𝒜 + ℬ in the set of squares 𝒮 (and, more generally, in the set of kth powers 𝒮k, where k ≥2 is an integer). It is known by a result of Gyarmati that 𝒜 + ℬ ⊂ 𝒮k ∩[1,N] implies that min(|𝒜|,|ℬ|) =Ok(logN). Here, we study how the upper bound on |ℬ| decreases, when the size of |𝒜| increases (or vice versa). In particular, if |𝒜| ≥ Ck1m m(logN)1m , then |ℬ| = Ok(m2logN), for sufficiently large N, a positive integer m and an explicit constant C > 0. For example, with m ∼ loglogN this gives: If |𝒜| ≥ CkloglogN,then |ℬ| = Ok(logN(loglogN)2).}, author = {Elsholtz, Christian and Wurzinger, Lena}, issn = {1464-3847}, journal = {The Quarterly Journal of Mathematics}, number = {4}, pages = {1243--1254}, publisher = {Oxford University Press}, title = {{Sumsets in the set of squares}}, doi = {10.1093/qmath/haae044}, volume = {75}, year = {2024}, } @article{15312, abstract = {The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.}, author = {Browning, Timothy D and Shparlinski, Igor E.}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {220--240}, publisher = {Elsevier}, title = {{Square-free values of random polynomials}}, doi = {10.1016/j.jnt.2024.02.013}, volume = {261}, year = {2024}, } @article{19051, abstract = {This paper corrects an error in an earlier work of the author.}, author = {Browning, Timothy D}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {13}, pages = {10165--10168}, publisher = {Oxford University Press}, title = {{The polynomial sieve and equal sums of like polynomials}}, doi = {10.1093/imrn/rnae066}, volume = {2024}, year = {2024}, } @article{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}, issn = {1475-3030 }, journal = {Journal of the Institute of Mathematics of Jussieu}, keywords = {Integral points, del Pezzo surface, universal torsor, Manin’s conjecture}, number = {3}, pages = {1259--1294}, publisher = {Cambridge University Press}, title = {{Integral points on singular del Pezzo surfaces}}, doi = {10.1017/S1474748022000482}, volume = {23}, year = {2024}, }