24 research outputs found

### Algebraic $S$-integers of fixed degree and bounded height

Let $k$ be a number field and $S$ a finite set of places of $k$ containing
the archimedean ones. We count the number of algebraic points of bounded height
whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an
asymptotic formula for the number of $\bar{S}$-integers of bounded height and
fixed degree over $k$, where $\bar{S}$ is the set of places of $\bar{k}$ lying
above the ones in $S$.Comment: arXiv admin note: text overlap with arXiv:1305.0482, accepted for
publication on Acta Arithmetic

### Torsion points with multiplicatively dependent coordinates on elliptic curves

In this paper, we study the finiteness problem of torsion points on an
elliptic curve whose coordinates satisfy some multiplicative dependence
relations. In particular, we prove that on an elliptic curve defined over a
number field there are only finitely many torsion points whose coordinates are
multiplicatively dependent. Moreover, we produce an effective result when the
elliptic curve is defined over the rational numbers or has complex
multiplication.Comment: 11 page

### Unlikely intersections in products of families of elliptic curves and the multiplicative group

Let $E_\lambda$ be the Legendre elliptic curve of equation
$Y^2=X(X-1)(X-\lambda)$. We recently proved that, given $n$ linearly
independent points $P_1(\lambda), \dots,P_n(\lambda)$ on $E_\lambda$ with
coordinates in $\bar{\mathbb{Q}(\lambda)}$, there are at most finitely many
complex numbers $\lambda_0$ such that the points $P_1(\lambda_0),
\dots,P_n(\lambda_0)$ satisfy two independent relations on $E_{\lambda_0}$. In
this article we continue our investigations on Unlikely Intersections in
families of abelian varieties and consider the case of a curve in a product of
two non-isogenous families of elliptic curves and in a family of split
semi-abelian varieties.Comment: To appear in The Quarterly Journal of Mathematic

### Additive unit representations in global fields - A survey

We give an overview on recent results concerning additive unit
representations. Furthermore the solutions of some open questions are included.
The central problem is whether and how certain rings are (additively) generated
by their units. This has been investigated for several types of rings related
to global fields, most importantly rings of algebraic integers. We also state
some open problems and conjectures which we consider to be important in this
field.Comment: 13 page

### Linear relations in families of powers of elliptic curves

Motivated by recent work of Masser and Zannier on simultaneous torsion on the
Legendre elliptic curve $E_\lambda$ of equation $Y^2=X(X-1)(X-\lambda)$, we
prove that, given $n$ linearly independent points $P_1(\lambda),
...,P_n(\lambda)$ on $E_\lambda$ with coordinates in
$\bar{\mathbb{Q}(\lambda)}$, there are at most finitely many complex numbers
$\lambda_0$ such that the points $P_1(\lambda_0), ...,P_n(\lambda_0)$ satisfy
two independent relations on $E_{\lambda_0}$. This is a special case of
conjectures about Unlikely Intersections on families of abelian varieties

### Unlikely Intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and
consider an abelian scheme A over S and a curve C inside A, both defined over
k. In previous works, we proved that when A is a fibered product of elliptic
schemes, if C is not contained in a proper subgroup scheme of A, then it
contains at most finitely many points that belong to a flat subgroup scheme of
codimension at least 2. In this article, we continue our investigation and
settle the crucial case of powers of simple abelian schemes of relative
dimension g bigger or equal than 2. This, combined with the above mentioned
result and work by Habegger and Pila, gives the statement for general abelian
schemes. These results have applications in the study of solvability of
almost-Pell equations in polynomials.Comment: 27 pages. Comments are welcome

### Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

Let $G$ be a semiabelian variety and $C$ a curve in $G$ that is not contained
in a proper algebraic subgroup of $G$. In this situation, conjectures of Pink
and Zilber imply that there are at most finitely many points contained in the
so-called unlikely intersections of $C$ with subgroups of codimension at least
$2$. In this note, we establish this assertion for general semiabelian
varieties over $\bar{\mathbb{Q}}$. This extends results of Maurin and Bombieri,
Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in
the abelian case.Comment: Comments are welcom

### CM relations in fibered powers of elliptic families

Let $E_\lambda$ be the Legendre family of elliptic curves. Given $n$ linearly
independent points $P_1,\dots , P_n \in
E_\lambda\left(\overline{\mathbb{Q}(\lambda)}\right)$ we prove that there are
at most finitely many complex numbers $\lambda_0$ such that $E_{\lambda_0}$
has complex multiplication and $P_1(\lambda_0), \dots ,P_n(\lambda_0)$ are
dependent over $End(E_{\lambda_0})$. This implies a positive answer to a
question of Bertrand and, combined with a previous work in collaboration with
Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an
elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.Comment: The formulation of Theorem 2.1 is now correc

### Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes

For positive integers $K$ and $L$, we introduce and study the notion of
$K$-multiplicative dependence over the algebraic closure
$\overline{\mathbb{F}}_p$ of a finite prime field $\mathbb{F}_p$, as well as
$L$-linear dependence of points on elliptic curves in reduction modulo primes.
One of our main results shows that, given non-zero rational functions
$\varphi_1,\ldots,\varphi_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an
elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently
large prime $p$, for all but finitely many $\alpha\in\overline{\mathbb{F}}_p$,
at most one of the following two can happen:
$\varphi_1(\alpha),\ldots,\varphi_m(\alpha)$ are $K$-multiplicatively dependent
or the points $(\varrho_1(\alpha),\cdot), \ldots,(\varrho_n(\alpha),\cdot)$ are
$L$-linearly dependent on the reduction of $E$ modulo $p$. As one of our main
tools, we prove a general statement about the intersection of an irreducible
curve in the split semiabelian variety $\mathbb{G}_{\mathrm{m}}^m \times E^n$
with the algebraic subgroups of codimension at least $2$.
As an application of our results, we improve a result of M. C. Chang and
extend a result of J. F. Voloch about elements of large order in finite fields
in some special cases.Comment: 32 pages. To appear in International Mathematics Research Notice

### Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes

For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure of a finite prime field Fp, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions Ï†1,â€¦,Ï†m,Ï±1,â€¦,Ï±n âˆˆ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many Î± in the algebraic closure of F_p, at most one of the following two can happen: Ï†1(Î±),â€¦,Ï†m(Î±) are K-multiplicatively dependent or the points (Ï±1(Î±),â‹…),â€¦,(Ï±n(Î±),â‹…) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety G^k_mÃ—E^n with the algebraic subgroups of codimension at least 2.
As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases