1,720,973 research outputs found
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
CM relations in fibered powers of elliptic families.
Get PDFLet (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.)
Unlikely intersections in products of families of elliptic curves and the multiplicative group
Get PDFLet 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
Linear relations in families of powers of elliptic curves
Get PDFMotivated 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
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
Additive unit representations in rings over global fields - a survey
No full textWe give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background
- …
