TY - JOUR AB - 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 $\ell$. AU - Lombardo, Davide AU - Verzobio, Matteo ID - 12312 IS - 2 JF - Selecta Mathematica SN - 1022-1824 TI - On the local-global principle for isogenies of abelian surfaces VL - 30 ER - TY - JOUR AB - 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. AU - Naskręcki, Bartosz AU - Verzobio, Matteo ID - 12311 JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics KW - Elliptic curves KW - Néron models KW - division polynomials KW - height functions KW - discrete valuation rings SN - 0308-2105 TI - Common valuations of division polynomials ER - TY - JOUR AB - We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split quadric surface AU - Browning, Timothy D AU - Lyczak, Julian AU - Sarapin, Roman ID - 13180 IS - 2 JF - Involve SN - 1944-4176 TI - Local solubility for a family of quadrics over a split quadric surface VL - 16 ER - TY - JOUR AB - 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. AU - Wilsch, Florian Alexander ID - 9034 IS - 8 JF - International Mathematics Research Notices SN - 1073-7928 TI - Integral points of bounded height on a log Fano threefold VL - 2023 ER - TY - JOUR AB - 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. AU - Balestrieri, Francesca ID - 12427 IS - 3 JF - Proceedings of the American Mathematical Society SN - 0002-9939 TI - Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups VL - 151 ER - TY - JOUR AB - 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. AU - Browning, Timothy D AU - Sawin, Will ID - 13091 IS - 3 JF - Algebra and Number Theory SN - 1937-0652 TI - Free rational curves on low degree hypersurfaces and the circle method VL - 17 ER - TY - JOUR 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 IS - 3 JF - Annals of Mathematics SN - 0003-486X TI - The Hasse principle for random Fano hypersurfaces VL - 197 ER - TY - JOUR AB - 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. AU - Bonolis, Dante AU - Browning, Timothy D ID - 12916 IS - 1 JF - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze SN - 0391-173X TI - Uniform bounds for rational points on hyperelliptic fibrations VL - 24 ER - TY - JOUR AB - 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. AU - Verzobio, Matteo ID - 12313 IS - 2 JF - Pacific Journal of Mathematics TI - Some effectivity results for primitive divisors of elliptic divisibility sequences VL - 325 ER - TY - JOUR AB - 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. AU - Lyczak, Julian ID - 13973 IS - 2 JF - Annales de l'Institut Fourier SN - 0373-0956 TI - Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces VL - 73 ER - TY - JOUR AB - We establish effective counting results for lattice points in families of domains in real, complex and quaternionic hyperbolic spaces of any dimension. The domains we focus on are defined as product sets with respect to an Iwasawa decomposition. Several natural diophantine problems can be reduced to counting lattice points in such domains. These include equidistribution of the ratio of the length of the shortest solution (x,y) to the gcd equation bx−ay=1 relative to the length of (a,b), where (a,b) ranges over primitive vectors in a disc whose radius increases, the natural analog of this problem in imaginary quadratic number fields, as well as equidistribution of integral solutions to the diophantine equation defined by an integral Lorentz form in three or more variables. We establish an effective rate of convergence for these equidistribution problems, depending on the size of the spectral gap associated with a suitable lattice subgroup in the isometry group of the relevant hyperbolic space. The main result underlying our discussion amounts to establishing effective joint equidistribution for the horospherical component and the radial component in the Iwasawa decomposition of lattice elements. AU - Horesh, Tal AU - Nevo, Amos ID - 14245 IS - 2 JF - Pacific Journal of Mathematics SN - 0030-8730 TI - Horospherical coordinates of lattice points in hyperbolic spaces: Effective counting and equidistribution VL - 324 ER - TY - JOUR AB - We count primitive lattices of rank d inside Zn as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subspace that a lattice spans, namely its projection to the Grassmannian; its homothety class and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude the joint equidistribution of these parameters. In addition to the primitive d-lattices Λ themselves, we also consider their orthogonal complements in Zn⁠, A1⁠, and show that the equidistribution occurs jointly for Λ and A1⁠. Finally, our asymptotic formulas for the number of primitive lattices include an explicit bound on the error term. AU - Horesh, Tal AU - Karasik, Yakov ID - 14717 IS - 4 JF - Quarterly Journal of Mathematics SN - 0033-5606 TI - Equidistribution of primitive lattices in ℝn VL - 74 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 - 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 - JOUR 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 IS - 10 JF - Algebra & Number Theory SN - 1937-0652 TI - Equidistribution and freeness on Grassmannians VL - 16 ER - TY - JOUR AB - Let t : Fp → C be a complex valued function on Fp. A classical problem in analytic number theory is bounding the maximum M(t) := max 0≤H

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 - 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 - 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 - 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 - Given a place ω of a global function field K over a finite field, with associated affine function ring Rω and completion Kω , the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points (a,b)∈Rω2 in the plane Kω2 , and for renormalized solutions to the gcd equation ax+by=1 . The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in \ZZ2 . AU - Horesh, Tal AU - Paulin, Frédéric ID - 12684 IS - 3 JF - Journal de Theorie des Nombres de Bordeaux SN - 1246-7405 TI - Effective equidistribution of lattice points in positive characteristic VL - 34 ER -