arXiv subjects

Umberto Zannier

Publications and source records attributed to Umberto Zannier.

At least 19 recordsLinked to original sources

arXiv2026

Pencils of norm form equations and a conjecture of Thomas, II

We continue our studies on parametric norm forms $F_t({\bf x})$, with ${\bf x}=(x_0,x_1,\ldots,x_{d-1})$ lying in some parametric linear subvariety $W_t$ and integers $t$ sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions $\bf x$ of $F_t({\bf x})=1$. Here we modify our techniques to treat $F_t({\bf x})=q$ for an arbitrary integer $q$. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all $\bf x$ are polynomially bounded in terms of $|q|$ and $t$. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with $x_2=\cdots=x_{d-1}=0$. We present an example with eight variables.

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arXiv2026

Rational and integral values of rational functions at rational points

The basic issue concerns sets of values of rational functions at rational points of an algebraic variety, namely image f(X(k)), where X is an algebbaric variety and f is a rational function on X, defined over the number field k. For instance, we shall prove that for X an abelian variety, the map between rational points is never surjective. This is reminiscent of the Hilbert Property, but here the fibers may have arbitrary dimension. One of our examples concerns the classical Hilbert Property: we produce a simply connected affine surface whose set of integral points is Zariski-dense and thin, disproving a plausible expectation. We shall also discuss hieghts and integrality issues; in this context, a role will be played by 'gcd estimates'. In the first Appendix, written by D. Masser, an effective estimate of some relevant gcd is provided.

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arXiv2026

Finding the complement of an elliptic curve inside a Jacobian

This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $π:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $π':X\to E'$ independent of $π$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here.

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arXiv2025

On the Disk of Convergence of Algebraic Power Series

This paper is mainly concerned with the disk of convergence of a power series s(x) representing an algebraic function of x and specifically with the relation between this disk and the branch points of the function. We shall focus especially on the p-adic case, answering some questions of basic nature, seemingly absent from the existing literature. Our methods are simple and essentially self-contained. To illustrate the issues, recall that in the complex case it follows from standard arguments that the open disk of convergence cannot contain all the branch points of x unless the series represents a rational function. In the p-adic case, we show that the analogous assertion is not true in complete generality; but we also confirm it in a number of cases, for instance under the assumption that p is not smaller than the degree of s(x) over the field of rational functions of x. In particular this gives an upper bound for the radius of convergence which has intrinsic nature. We shall also touch several related questions.

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arXiv2025

A problem of Polya

We prove an improved form of an expectation of Polya and discuss several related questions

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arXiv2024

Examples of effectivity for integral points on certain curves of genus 2

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively determined. Our method is based on the construction of degree-3 étale covers of such curves of genus 2 and the study of the torsion values of sections of certain doubly elliptic abelian schemes.

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arXiv2024

Rational distances from given rational points in the plane

In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in the real topology. We show in the Main Theorem that these conditions can be checked effectively (via congruences), proving that a related class of K3 surfaces satisfies the local-global principle. In particular, these conditions are always satisfied when P_1, P_2, P_3 are rational. This result completes and goes beyond the analysis of Berry, who worked under stronger assumptions, not always fulfilled for instance in all the cases where P_1, P_2, P_3 are rational. On the other hand, for r\ge 4, we show that points with rational distances correspond to rational points in a surface of general type, hence conjecturally not Zariski dense. However, at the present, we lack methods to prove this, given the fact that the surface is simply-connected, as we shall show. We give explicit proofs as well as describe in detail the geometry of the surfaces involved. In addition we discuss certain analogues for points with distances in certain ring of integers.

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arXiv2024

Classification of rational angles in plane lattices

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $π$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathematical Society

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arXiv2024

Classification of rational angles in plane lattices II

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of $π$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

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arXiv2023

Cyclotomic and abelian points in backward orbits of rational functions

We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $ϕ\in K(x)$ and $α\in K$ then the extension of $K$ generated by the abelian points in the backward orbit of $α$ is ramified only at finitely many primes. This has the immediate strong consequence that if all points in the backward orbit of $α$ are abelian then $ϕ$ is post-critically finite. We use this result to prove two facts: on the one hand, if $ϕ\in \mathbb Q(x)$ is a quadratic rational function not conjugate over $\mathbb Q^{\text{ab}}$ to a power or a Chebyshev map and all preimages of $α$ are abelian, we show that $ϕ$ is $\mathbb Q$-conjugate to one of two specific quadratic functions, in the spirit of a recent conjecture of Andrews and Petsche. On the other hand we provide conditions on a quadratic rational function in $K(x)$ for the backward orbit of a point $α$ to only contain finitely many cyclotomic preimages, extending previous results of the second author. Finally, we give necessary and sufficient conditions for a triple $(ϕ,K,α)$, where $ϕ$ is a Lattès map over a number field $K$ and $α\in K$ for the whole backward orbit of $α$ to only contain abelian points.

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arXiv2023

Purely exponential parametrizations and their group-theoretic applications

This paper is mainly motivated by the analysis of the so-called Bounded Generation property (BG) of linear groups (in characteristic $0$), which is known to admit far-reaching group-theoretic implications. We achieve complete answers to certain longstanding open questions about Bounded Generation (sharpening considerably some earlier results). For instance, we prove that linear groups boundedly generated by semi-simple elements are necessarily virtually abelian. This is obtained as a corollary of sparseness of subsets which are likewise generated. In the paper in fact we go further, framing (BG) in the more general context of (Purely) Exponential Parametrizations (PEP) for subsets of affine spaces, a concept which unifies different issues. Using deep tools from Diophantine Geometry (including the Subspace Theorem), we systematically develop a theory showing in particular that for a (PEP) set over a number field, the asymptotic distribution of its points of Height at most $T$ is always $\sim c(\log T)^r$, with certain constants $c>0$ and $r\in \mathbb{Z}_{\geq 0}$. (This shape fits with a well-known viewpoint first put forward by Manin.)

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arXiv2023

Examples of effectivity for integral points on certain curves of genus 2

This short article concerns a method to obtain effectivity for the search of integral points on certain (sets of) curves of genus 2. More precisely, we wish to illustrate just an example of application of a criterion of Bilu, to derive effectivity for integral points on certain families of affine curves of genus 2. Future work, in collaboration with D. Lombardo, will contain details and more general applications. We shall construct morphisms from these curves to $\G_{\rm m}^2$, with image of increasing degrees. We note that as the degree increases, we may say that the examples become `more interesting', since they cannot be derived by substitution from a universal family. As a counterpart, there is the negative feature in that the relevant curves will somewhat have increasing fields of definition.

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arXiv2023

D-finiteness, rationality, and height III: multivariate Pólya-Carlson dichotomy

We prove a result that can be seen as an analogue of the Pólya-Carlson theorem for multivariate D-finite power series with coefficients in $\bar{\mathbb{Q}}$. In the special case that the coefficients are algebraic integers, our main result says that if $$F(x_1,\ldots ,x_m)=\sum f(n_1,\ldots ,n_m)x_1^{n_1}\cdots x_m^{n_m}$$ is a D-finite power series in $m$ variables with algebraic integer coefficients and if the logarithmic Weil height of $f(n_1,\ldots ,n_m)$ is $o(n_1+\cdots +n_m)$, then $F$ is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of $F$ has the form $1-ζx_1^{q_1}\cdots x_m^{q_m}$ where $ζ$ is a root of unity and $q_1,\ldots ,q_m$ are nonnegative integers, not all of which are zero.

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arXiv2023

Finite Orbits in Surfaces with a Double Elliptic Fibration and Torsion Values of Sections

We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a similar direction of (and is motivated by) recent work by Cantat-Dujardin, although we use very different methods and obtain related but different results. As a sample of application of similar arguments, we prove a new case of the Zilber-Pink conjecture, namely Theorem 1.5, for certain schemes over a 2-dimensional base, which was known to lead to substantial difficulties. Most results rely, among other things, on recent theorems by Bakker and the second author of `Ax-Schanuel Type'; we also relate a functional condition with a theorem of Shioda on ramified sections of the Legendre scheme. For one of our proofs, we also use recent height inequalities by Yuan-Zhang. Finally, in an appendix, we show that the Relative Manin-Mumford Conjecture over the complex number field is equivalent to its version over the field of algebraic numbers.

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arXiv2022

D-finiteness, rationality, and height II: lower bounds over a set of positive density

We consider D-finite power series $f(z)=\sum a_n z^n$ with coefficients in a number field $K$. We show that there is a dichotomy governing the behaviour of $h(a_n)$ as a function of $n$, where $h$ is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either $f(z)$ is rational or $h(a_n)>[K:\mathbb{Q}]^{-1}\cdot \log(n)+O(1)$ for $n$ in a set of positive upper density and this is best possible when $K=\mathbb{Q}$.

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arXiv2022

On the distribution of rational points on ramified covers of abelian varieties

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $π: X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $π(X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the Inverse Galois Problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.

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arXiv2022

Bounded Generation by semi-simple elements: quantitative results

We prove that for a number field $F$, the distribution of the points of a set $Σ\subset \mathbb{A}_F^n$ with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group $Γ\subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal $m$-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse's strengthening of the $S$-Unit Equation Theorem.

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arXiv2021

Betti maps, Pell equation in polynomials and almost Belyi maps

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$, with $A,B,D\in \mathbb C[t]$ and certain ramified covers ${\mathbb P}^1\to {\mathbb P}^1$ arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Covaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials $D$ that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann Existence Theorem associates to the above-mentioned covers certain permutation representations: we are able to characterize the representations corresponding to "primitive" solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when $D$ has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.

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