192 research outputs found

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

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    Let EλE_\lambda be the Legendre elliptic curve of equation Y2=X(X−1)(X−λ)Y^2=X(X-1)(X-\lambda). We recently proved that, given nn linearly independent points P1(λ),…,Pn(λ)P_1(\lambda), \dots,P_n(\lambda) on EλE_\lambda with coordinates in Q(λ)ˉ\bar{\mathbb{Q}(\lambda)}, there are at most finitely many complex numbers λ0\lambda_0 such that the points P1(λ0),…,Pn(λ0)P_1(\lambda_0), \dots,P_n(\lambda_0) satisfy two independent relations on Eλ0E_{\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

    Rational points on Grassmannians and unlikely intersections in tori

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    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

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    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

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    Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve EλE_\lambda of equation Y2=X(X−1)(X−λ)Y^2=X(X-1)(X-\lambda), we prove that, given nn linearly independent points P1(λ),...,Pn(λ)P_1(\lambda), ...,P_n(\lambda) on EλE_\lambda with coordinates in Q(λ)ˉ\bar{\mathbb{Q}(\lambda)}, there are at most finitely many complex numbers λ0\lambda_0 such that the points P1(λ0),...,Pn(λ0)P_1(\lambda_0), ...,P_n(\lambda_0) satisfy two independent relations on Eλ0E_{\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

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    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 n≥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

    A note on cyclotomic polynomials and Linear Feedback Shift Registers

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    Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then f(an)=0f(a^n)=0. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases K=FpK=\mathbb{F}_p and K=QK=\mathbb{Q}.Comment: 12 page

    A note on cyclotomic polynomials and Linear Feedback Shift Registers

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    Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then f(a^n)=0. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases K=Fp and K=Q
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