@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{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}, publisher = {Cambridge University Press}, title = {{Common valuations of division polynomials}}, doi = {10.1017/prm.2024.7}, year = {2024}, } @article{13180, abstract = {We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split quadric surface}, author = {Browning, Timothy D and Lyczak, Julian and Sarapin, Roman}, issn = {1944-4184}, journal = {Involve}, number = {2}, pages = {331--342}, publisher = {Mathematical Sciences Publishers}, title = {{Local solubility for a family of quadrics over a split quadric surface}}, doi = {10.2140/involve.2023.16.331}, volume = {16}, year = {2023}, } @article{9034, abstract = {We determine an asymptotic formula for the number of integral points of bounded height on a blow-up of P3 outside certain planes using universal torsors.}, author = {Wilsch, Florian Alexander}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {8}, pages = {6780--6808}, publisher = {Oxford Academic}, title = {{Integral points of bounded height on a log Fano threefold}}, doi = {10.1093/imrn/rnac048}, volume = {2023}, year = {2023}, } @article{12427, abstract = {Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g : Z → X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1(k, G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent.}, author = {Balestrieri, Francesca}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {3}, pages = {907--914}, publisher = {American Mathematical Society}, title = {{Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups}}, doi = {10.1090/proc/15239}, volume = {151}, year = {2023}, } @article{13091, abstract = {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.}, author = {Browning, Timothy D and Sawin, Will}, issn = {1944-7833}, journal = {Algebra and Number Theory}, number = {3}, pages = {719--748}, publisher = {Mathematical Sciences Publishers}, title = {{Free rational curves on low degree hypersurfaces and the circle method}}, doi = {10.2140/ant.2023.17.719}, volume = {17}, year = {2023}, } @article{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}, issn = {0003-486X}, journal = {Annals of Mathematics}, number = {3}, pages = {1115--1203}, publisher = {Princeton University}, title = {{The Hasse principle for random Fano hypersurfaces}}, doi = {10.4007/annals.2023.197.3.3}, volume = {197}, year = {2023}, } @article{12916, abstract = {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. }, author = {Bonolis, Dante and Browning, Timothy D}, issn = {2036-2145}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, number = {1}, pages = {173--204}, publisher = {Scuola Normale Superiore - Edizioni della Normale}, title = {{Uniform bounds for rational points on hyperelliptic fibrations}}, doi = {10.2422/2036-2145.202010_018}, volume = {24}, year = {2023}, } @article{12313, abstract = {Let P be a nontorsion point on an elliptic curve defined over a number field K and consider the sequence {Bn}n∈N of the denominators of x(nP). We prove that every term of the sequence of the Bn has a primitive divisor for n greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.}, author = {Verzobio, Matteo}, issn = {0030-8730}, journal = {Pacific Journal of Mathematics}, number = {2}, pages = {331--351}, publisher = {Mathematical Sciences Publishers}, title = {{Some effectivity results for primitive divisors of elliptic divisibility sequences}}, doi = {10.2140/pjm.2023.325.331}, volume = {325}, year = {2023}, } @article{13973, abstract = {We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer–Manin obstruction to the integral Hasse principle.}, author = {Lyczak, Julian}, issn = {0373-0956}, journal = {Annales de l'Institut Fourier}, number = {2}, pages = {447--478}, publisher = {Association des Annales de l'Institut Fourier}, title = {{Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces}}, doi = {10.5802/aif.3529}, volume = {73}, year = {2023}, }