On Mordell-Weil groups of elliptic curves induced by Diophantine triples

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.Comment: 17 pages; to appear in Glasnik Matematicki 42 (2007

Modular curves over number fields and ECM

International audienceWe construct families of elliptic curves defined over number fields and containing torsion groups Z=M1Z x Z=M2Z where (M1;M2) belongs to f(1; 11), (1; 14), (1; 15), (2; 10), (2; 12), (3; 9), (4; 8), (6; 6)g (i.e., when the corresponding modular curve X1(M1;M2) has genus 1). We provide formulae for the curves and give examples of number fields for which the corresponding elliptic curves have non-zero ranks, giving explicit generators using D. Simon's program whenever possible. The reductions of these curves can be used to speed up ECM for factoring numbers with special properties, a typical example being (factors of) Cunningham numbers bn - 1 such that M1 j n. We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists

A geometric approach to elliptic curves with torsion groups Z/10Z, Z/12Z, Z/14Z, and Z/16Z

We give new parametrisations of elliptic curves in Weierstrass normal form y2 = x3 + ax2 + bx with torsion groups Z/10Z and Z/12Z over Q, and with Z/14Z and Z/16Z over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group Z/12Z and positive rank. Furthermore, we found elliptic curves with torsion group Z/14Z and rank 3 - which is a new record for such curves - as well as some new elliptic curves with torsion group Z/16Z and rank 3

Locally imprimitive points on elliptic curves

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that are known to have infinitely many primes $\mathfrak p$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak p})$. We describe this phenomenon in terms of an associated Galois representation $\rho_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.Comment: 20 page