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Umberto Zannier

Publications and source records attributed to Umberto Zannier.

At least 37 records · Page 2Linked to original sources
arXiv2021

Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture

We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in $(0,π)$, there are only finitely many cases for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results have their origin in `relative' cases of the Manin-Mumford conjecture, and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell-Lang conjecture. In turn, this embraces cases which, somewhat surprisingly, can be treated (only) by completely different methods compared to the former; here we shall offer an explicit example related to diophantine equations in algebraic tori.

math.NT↗
arXiv2021

Non-virtually abelian anisotropic linear groups are not boundedly generated

We prove that if a linear group $Γ\subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite $S$-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are never boundedly generated. Our proof relies on Laurent's theorem from Diophantine geometry and properties of generic elements.

math.GR↗
arXiv2021

Around the Chevalley-Weil Theorem

We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.

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arXiv2021

Integral distances from (two) lattice points

{\it .}We completely characterize pairs of lattice points $P_1\neq P_2$ in the plane with the property that there are infinitely many lattice points $Q$ whose distance from both $P_1$ and $P_2$ is integral. In particular we show that it suffices that $P_2-P_1\neq (\pm 1,\pm 2), (\pm 2,\pm 1)$, and we show that $|P_1-P_2|>\sqrt{20}$ suffices for having infinitely many such $Q$ outside any finite union of lines. We use only elementary arguments, the crucial ingredient being a theorem of Gauss which does not appear to be often applied. We further include related remarks (and open questions), also for distances from an arbitrary prescribed finite set of lattice points % $P_1,\ldots ,P_r$. }

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arXiv2020

A note on the zeroes of the Fredholm series

The issue had been raised whether the Fredholm series $z+z^2+...+z^{2^n}+...$ has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times.

math.CV↗
arXiv2020

Analytic and rational sections of relative semi-abelian varieties

The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme over an affine algebraic curve. These sections form a group; while the group of rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

math.CV↗
arXiv2019

On the torsion values for sections of an elliptic scheme

We shall consider sections of an elliptic scheme $\mathcal{E}$ over a(n affine) base curve $B$, and study the points of $B$ where the section takes a torsion value. In particular, we shall relate the distribution in $B$ of these points with the canonical height of the section, proving an integral formula involving a measure on $B$ coming from the so-called Betti map of the section. We shall show that this measure is the same appearing in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on $\mathcal{E}$ (over the affine ring of $B$), and in the last part of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel's theorem on integral points.

math.AG↗
arXiv2019

Fibred algebraic surfaces and commutators in the Symplectic group

We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.

math.AG↗
arXiv2019

A variant of the Mordell-Lang conjecture

The Mordell-Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semiabelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $Γ\le G(\mathbb{k})$ is a finite union of cosets of subgroups of $Γ$. We explore a variant of this conjecture when $G$ is a product of an abelian variety $A$ defined over $\mathbb{k}$ with the additive group $\mathbb{G}_a$.

math.NT↗
arXiv2019

D-finiteness, rationality, and height

Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic 0 is D-finite and its coefficients belong to a finite set then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series `look like' the coefficients of a rational function then the power series is rational. Our work relies on the theory of Weil heights, the Manin-Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.

math.NT↗
arXiv2018

An effective criterion for periodicity of l-adic continued fractions

The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the l-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion, moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol).

math.NT↗
arXiv2018

On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem

In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a broad generalisation of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of initial points with two multiplicatively dependent elements in their orbits.

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arXiv2018

The Dynamical Mordell-Lang Conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of all positive integers n for which f^n(x) lies on V is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets consisting of all integers of the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..,n_m among all positive integers), for some given positive integer m, some rational numbers c_i and some non-negative integers k_i. We prove that this conjecture is equivalent with some difficult Diophantine problem in characteristic 0. In the case X is an algebraic torus, we can prove the conjecture in two cases: either when the dimension of V is at most 2, or when no iterate of f is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of X.

math.NT↗
arXiv2018

The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao)

Given a point $ξ$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $ξ$. When $(A, ξ)$ varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often $ξ$ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when $ξ$ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to $(A, ξ)$ (assuming $A$ without fixed part, and $\mathbb{Z} ξ$ Zariski-dense in $A$), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension $\leq 3$, and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if $A\to S$ is a principally polarized abelian scheme of relative dimension $g$ which has no non-trivial endomorphism (on any finite covering), and if the image of $S$ in the moduli space $\mathcal{A}_g$ has dimension at least $g$, then the Betti map of any non-torsion section $ξ$ is generically a submersion, so that $ξ^{-1}A_{tors}$ is dense in $S(\mathbb{C})$.

math.AG↗
arXiv2018

On abelian multiplicatively dependent points on a curve in a torus

We show, under some natural conditions, that the set of abelian (and thus also cyclotomic) multiplicatively dependent points on an irreducible curve over a number field is a finite union of preimages of roots of unity by a certain finite set of primitive characters from $\Gm^n$ to $\Gm$ restricted to the curve, and a finite set. We also introduce the notion of primitive multiplicative dependence and obtain a finiteness result for primitively multiplicatively dependent points defined over a so-called Bogomolov extension of a number field.

math.NT↗
arXiv2017

On fewnomials, integral points and a toric version of Bertini's theorem

An old conjecture of Erdős and Rényi, proved by Schinzel, predicted a bound for the number of terms of a polynomial $g(x) \in \mathbb{C}[x]$ when its square $g(x)^2$ has a given number of terms. Further conjectures and results arose, but some fundamental questions remained open. In this paper, with methods which appear to be new, we achieve a final result in this direction for completely general algebraic equations $f(x,g(x))=0$, where $f(x,y)$ is monic of arbitrary degree in $y$, and has boundedly many terms in $x$: we prove that the number of terms of such a $g(x)$ is necessarily bounded. This includes the previous results as extremely special cases. We shall interpret polynomials with boundedly many terms as the restrictions to 1-parameter subgroups or cosets of regular functions of bounded degree on a given torus $\mathbb{G}_{\mathrm{m}}^l$. Such a viewpoint shall lead to some best-possible corollaries in the context of finite covers of $\mathbb{G}_{\mathrm{m}}^l$, concerning the structure of their integral points over function fields (in the spirit of conjectures of Vojta) and a Bertini-type irreducibility theorem above algebraic multiplicative cosets. A further natural reading occurs in non-standard arithmetic, where our result translates into an algebraic and integral-closedness statement inside the ring of non-standard polynomials.

math.NT↗
arXiv2016

On the Hilbert Property and the Fundamental Group of Algebraic Varieties

We review, under a perspective which appears different from previous ones, the so-called Hilbert Property (HP) for an algebraic variety (over a number field); this is linked to Hilbert's Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem. We shall observe that the HP is in a sense `opposite' to the Chevalley-Weil Theorem, which concerns unramified covers; this link shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the HP, involving Enriques surfaces. We also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a non-rational variety for which the HP can be proved. We also formulate some general conjectures relating the HP with the topology of algebraic varieties.

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