25 research outputs found
CM relations in fibered powers of elliptic families.
No full textLet (Formula presented.) be the Legendre family of elliptic curves. Given (Formula presented.) points (Formula presented.), linearly independent over (Formula presented.), we prove that there are at most finitely many complex numbers (Formula presented.) such that (Formula presented.) has complex multiplication and (Formula presented.) are linearly dependent over End(Formula presented.). 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 (Formula presented.)
Counting algebraic integers of fixed degree and bounded height
No full textLet k be a number field. For H→∞, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by H and fixed degree over k
Unlikely intersections in products of families of elliptic curves and the multiplicative group
No full textLet Eλbe the Legendre elliptic curve of equation Y2= X (X - 1)(X - l). We recently proved that, given n linearly independent points P1(l), 1⁄4, Pn(l) on Eλwith coordinates in (l), there are at most finitely many complex numbers l0such that the points P1(l0), 1⁄4, Pn(l0) satisfy two independent relations on El0. 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
TORSION POINTS WITH MULTIPLICATIVELY DEPENDENT COORDINATES ON ELLIPTIC CURVES
No full textIn 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
Counting Lattice Points and O-Minimal Structures
No full textLet Λ be a lattice in Rn, and let Z ⊆Rm+n be a definable family in an O-minimal structure over R. We give sharp estimates for the number of lattice points in the fibers ZT = {x∈ Rn: (T, x) ∈ Z}. Along the way, we show that for any subspace Σ ⊆Rn of dimension j> 0 the j-volume of the orthogonal projection of ZT to Σ is, up to a constant depending only on the family Z, bounded by the maximal j-dimensional volume of the orthogonal projections to the j-dimensional coordinate subspaces
On the Zilber-Pink conjecture for complex abelian varieties
No full textIn this article, we prove that the Zilber-Pink conjecture for abelian
varieties over an arbitrary field of characteristic is implied by the same
statement for abelian varieties over the algebraic numbers.
More precisely, the conjecture holds for subvarieties of dimension at most
in the abelian variety if it holds for subvarieties of dimension at
most in the largest abelian subvariety of that is isomorphic to an
abelian variety defined over
Linear relations in families of powers of elliptic curves
No full textMotivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve Eλof equation Y2= X (X − 1)(X − λ), we prove that, given n linearly independent points P1(λ),..., Pn(λ) on Eλwith coordinates in Q(λ), there are at most finitely many complex numbers λ0 such that the points P1(λ0),..., Pn(λ0) satisfy two independent relations on Eλ0. This is a special case of conjectures about unlikely intersections on families of abelian varieties
