An equivalence of categories in algebraic geometry and some unlikely intersections in powers of elliptic curves
This talk will be divided into two sections. In the first part, we introduce a definition of affine varieties, in the classical sense, over an arbitrary field k and describe two functors which give an equivalence of categories between the category of these affine varieties and reduced schemes of finite type over k. Without defining prevarieties that can be constructed using our introduced definition of classical affine varieties, this part particularly illuminates the basic ideas used in proving an equivalence of categories between the category of prevarieties over k and reduced schemes of finite type over k.
In the final part, we briefly discuss the proof, highlighting some constructions and model-theoretic ideas used, of the following result generalising work of Boxall:
Let E be an elliptic curve defined over a number field, and let C1, C2 ⊆ E^N(ℂ) be irreducible closed algebraic curves where N ≥ 3 and at least one of C1 and C2 is not defined over the algebraic numbers. Suppose that C1 is not contained in a 1-dimensional algebraic subgroup of E^N(ℂ) and C1 ∩ C2 is not contained in a 2-dimensional algebraic subgroup of E^N(ℂ). Then there are at most finitely many points x ∈ C1 such that there exists an n ∈ N such that nx ∈ C2 and that [n]C1 ⊈ C2 where [n]C1 = {nx : x ∈ C1}.