221 research outputs found

### 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

### Upper ramification jumps in abelian extensions of exponent p

In this paper we present a classification of the possible upper ramification
jumps for an elementary abelian p-extension of a p-adic field. The fundamental
step for the proof of the main result is the computation of the ramification
filtration for the maximal elementary abelian p-extension of the base field K.
This is a generalization of a previous work of the second author and Dvornicich
where the same result is proved under the assumption that K contains a
primitive p-th root of unity. Using the class field theory and the explicit
relations between the normic group of an extension and its ramification jumps,
it is fairly simple to recover necessary and sufficient conditions for the
upper ramification jumps of an elementary abelian p-extension of K.Comment: 9 page

### Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to
Bombieri, Masser and Zannier concerning intersections of a curve in the
multiplicative group of dimension n with algebraic subgroups of dimension n-2.
The proof uses a method introduced for the first time by Pila and Zannier to
give an alternative proof of Manin-Mumford conjecture and a theorem to count
points that satisfy a certain number of linear conditions with rational
coefficients. This method has been largely used in many different problems in
the context of "unlikely intersections".Comment: 16 page

### 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

### 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

### On periodicity of p-adic Browkin continued fractions

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every nâ‰¥1, there exist infinitely many square roots of integers with periodic Browkin expansion of period 2^n, extending a previous result of Bedocchi obtained for n=1

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