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On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0
Let K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor_0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor_0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.
Unlikely intersections with isogeny orbits in a product of elliptic schemes
Distinguished Categories and the Zilber-Pink Conjecture
We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic concepts of the field and to prove some fundamental facts about them, e.g., the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This yields unconditional results, i.e., the Zilber-Pink conjecture for a complex curve in $\mathcal{A}_2$ that cannot be defined over $\bar{\mathbb{Q}}$, a complex curve in the g-th fibered power of the Legendre family, and a complex curve in the base change of a semiabelian variety over $\bar{\mathbb{Q}}$.
On a Galois property of fields generated by the torsion of an abelian variety
In this article, we study a certain Galois property of subextensions of k(A_tors), the minimal field of definition of all torsion points of an abelian variety A defined over a number field k. Concretely, we show that each subfield of k(A_tors) that is Galois over k (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of k. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height.