@article{21002, abstract = {The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.}, author = {Browning, Timothy D}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{The Davenport–Heilbronn method: 80 years on}}, doi = {10.1112/jlms.70371}, volume = {113}, year = {2026}, } @article{21242, abstract = {We obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating recent progress on Waring’s problem and the resolution of the main conjecture in Vinogradov’s mean value theorem.}, author = {Rome, Nick and Yamagishi, Shuntaro}, issn = {1945-5844}, journal = {Pacific Journal of Mathematics}, number = {1}, pages = {179--198}, publisher = {Mathematical Sciences Publishers}, title = {{Integral solutions to systems of diagonal equations}}, doi = {10.2140/pjm.2026.340.179}, volume = {340}, year = {2026}, } @article{21385, abstract = {We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. }, author = {Wang, Victor and Xu, Max}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, pages = {1--15}, publisher = {Cambridge University Press}, title = {{Average sizes of mixed character sums}}, doi = {10.1017/prm.2026.10123}, year = {2026}, } @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{12311, abstract = {In this note, we prove a formula for the cancellation exponent kv,n between division polynomials ψn and ϕn associated with a sequence {nP}n∈N of points on an elliptic curve E defined over a discrete valuation field K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.}, author = {Naskręcki, Bartosz and Verzobio, Matteo}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, keywords = {Elliptic curves, Néron models, division polynomials, height functions, discrete valuation rings}, number = {5}, pages = {1646--1660}, publisher = {Cambridge University Press}, title = {{Common valuations of division polynomials}}, doi = {10.1017/prm.2024.7}, volume = {155}, 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{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{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{18822, abstract = {Let N(X) be the number of integral zeros (mathematical equation). Works of Hooley and Heath-Brown imply (mathematical equation), if one assumes automorphy and grand Riemann hypothesis for certain Hasse–Weil L-functions. Assuming instead a natural large sieve inequality, we recover the same bound on N(X). This is part of a more general statement, for diagonal cubic forms in (mathematical equation) variables, where we allow approximations to Hasse–Weil L-functions.}, author = {Wang, Victor}, issn = {2041-7942}, journal = {Mathematika}, number = {1}, publisher = {London Mathematical Society}, title = {{Diagonal cubic forms and the large sieve}}, doi = {10.1112/mtk.70008}, volume = {71}, year = {2025}, } @article{19054, abstract = {This work concerns asymptotical stabilisation phenomena occurring in the moduli space of sections of certain algebraic families over a smooth projective curve, whenever the generic fibre of the family is a smooth projective Fano variety, or not far from being Fano. We describe the expected behaviour of the class, in a ring of motivic integration, of the moduli space of sections of given numerical class. Up to an adequate normalisation, it should converge, when the class of the sections goes arbitrarily far from the boundary of the dual of the effective cone, to an effective element given by a motivic Euler product. Such a principle can be seen as an analogue for rational curves of the Batyrev-Manin-Peyre principle for rational points. The central tool of this article is the property of equidistribution of curves. We show that this notion does not depend on the choice of a model of the generic fibre, and that equidistribution of curves holds for smooth projective split toric varieties. As an application, we study the Batyrev-Manin-Peyre principle for curves on a certain kind of twisted products.}, author = {Faisant, Loïs}, issn = {1944-7833}, journal = {Algebra & Number Theory}, pages = {883--965}, publisher = {Mathematical Sciences Publishers}, title = {{Motivic distribution of rational curves and twisted products of toric varieties}}, doi = {10.2140/ant.2025.19.883}, volume = {19}, year = {2025}, } @unpublished{19055, abstract = {Using the formalism of Cox rings and universal torsors, we prove a decomposition of the Grothendieck motive of the moduli space of morphisms from an arbitrary smooth projective curve to a Mori Dream Space (MDS). For the simplest cases of MDS, that of toric varieties, we use this decomposition to prove an instance of the motivic Batyrev--Manin--Peyre principle for curves satisfying tangency conditions with respect to the boundary divisors, often called Campana curves.}, author = {Faisant, Loïs}, booktitle = {arXiv}, title = {{Motivic counting of rational curves with tangency conditions via universal torsors}}, doi = {10.48550/ARXIV.2502.11704}, year = {2025}, } @article{19363, abstract = {For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.}, author = {Chan, Yik Tung and Koymans, Peter and Pagano, Carlo and Sofos, Efthymios}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {1--36}, publisher = {Elsevier}, title = {{Averages of multiplicative functions along equidistributed sequences}}, doi = {10.1016/j.jnt.2025.01.005}, volume = {273}, 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{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{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{19776, abstract = {We use the circle method to prove that a density 1 of elements in Fq[t] are representable as a sum of three cubes of essentially minimal degree from Fq[t], assuming the Ratios Conjecture and that char(Fq)>3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.}, author = {Browning, Timothy D and Glas, Jakob and Wang, Victor}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {4}, publisher = {Springer Nature}, title = {{Optimal sums of three cubes in Fq[t]}}, doi = {10.1007/s00209-025-03765-z}, volume = {310}, year = {2025}, } @article{20850, abstract = {We provide an estimate for the number of nontrivial integer points on the Pellian surface t^2 - du^2 = 1 in a bounded region. We give a lower bound on the size of fundamental solutions for almost all d in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.}, author = {Diao, Yijie}, issn = {2118-8572}, journal = {Journal de theorie des nombres de Bordeaux}, number = {3}, pages = {973--988}, publisher = {Université de Bordeaux}, title = {{Class numbers and integer points on some Pellian surfaces}}, doi = {10.5802/jtnb.1348}, volume = {37}, year = {2025}, } @article{21003, abstract = {We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2397-3129}, journal = {Discrete Analysis}, publisher = {Cambridge: Alliance of Diamond Open Access Journals}, title = {{Counting integer points on affine surfaces with a side condition}}, doi = {10.19086/da.143787}, volume = {2025}, year = {2025}, }