24 research outputs found

    Algebraic SS-integers of fixed degree and bounded height

    Full text link
    Let kk be a number field and SS a finite set of places of kk containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of SS-integers of kk. Moreover, we give an asymptotic formula for the number of Sˉ\bar{S}-integers of bounded height and fixed degree over kk, where Sˉ\bar{S} is the set of places of kˉ\bar{k} lying above the ones in SS.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

    Full text link
    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

    Full text link
    Let EλE_\lambda be the Legendre elliptic curve of equation Y2=X(X1)(Xλ)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

    Additive unit representations in global fields - A survey

    Full text link
    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

    Full text link
    Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve EλE_\lambda of equation Y2=X(X1)(Xλ)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

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

    Full text link
    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

    Get PDF
    Let GG be a semiabelian variety and CC a curve in GG that is not contained in a proper algebraic subgroup of GG. 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 CC with subgroups of codimension at least 22. In this note, we establish this assertion for general semiabelian varieties over Qˉ\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

    Full text link
    Let EλE_\lambda be the Legendre family of elliptic curves. Given nn linearly independent points P1,,PnEλ(Q(λ))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 λ0\lambda_0 such that Eλ0E_{\lambda_0} has complex multiplication and P1(λ0),,Pn(λ0)P_1(\lambda_0), \dots ,P_n(\lambda_0) are dependent over End(Eλ0)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 Q\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

    Get PDF
    For positive integers KK and LL, we introduce and study the notion of KK-multiplicative dependence over the algebraic closure Fp\overline{\mathbb{F}}_p of a finite prime field Fp\mathbb{F}_p, as well as LL-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions φ1,,φm,ϱ1,,ϱnQ(X)\varphi_1,\ldots,\varphi_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X) and an elliptic curve EE defined over the integers Z\mathbb{Z}, for any sufficiently large prime pp, for all but finitely many αFp\alpha\in\overline{\mathbb{F}}_p, at most one of the following two can happen: φ1(α),,φm(α)\varphi_1(\alpha),\ldots,\varphi_m(\alpha) are KK-multiplicatively dependent or the points (ϱ1(α),),,(ϱn(α),)(\varrho_1(\alpha),\cdot), \ldots,(\varrho_n(\alpha),\cdot) are LL-linearly dependent on the reduction of EE modulo pp. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety Gmm×En\mathbb{G}_{\mathrm{m}}^m \times E^n with the algebraic subgroups of codimension at least 22. 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

    Get PDF
    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 φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) 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_m×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
    corecore