Titre : Relations on a power of an elliptic curve
(work in progress)

Résumé :

Let E be an elliptic curve and let X a subvariety of the algebraic group Eⁿ. We study the solutions of independent linear relations
a_i1p₁ + ... + a_inp_n = 0     (p₁,...,p_n)∈ X
with a_ij elements of the endomorphism ring of E. The a_ij are to be considered as varying and the number of relations will be fixed and a function of dim X. Equivalently one could study the intersection of X with the union of all algebraic subgroups of fixed dimension. The following conjecture has been stated independently and sometimes in somewhat different form by several authors ([Bombieri, Masser, Zannier] and [Pink] and [Zilber]): say X is a subvariety of a semi-abelian variety but not contained in a proper algebraic subgroup, then the intersection of X with the union of all algebraic subgroups of dimension at most n-dim X-1 is not Zariski dense in X. We discuss a result which proves this conjecture under a stronger, geometric hypothesis on X and if A=Eⁿ and E has complex multiplication. The proof involves the theory of heights from Diophantine Geometry.

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