Connectedness extensions for abelian varieties (Q1127802)

Let \(A\) be an abelian variety over a field \(F\). There is a minimal extension \(F'(A)\) of \(F\) such that the algebraic closure of the image of the Galois group \(\text{Gal} (F^{\text{sep}}/F')\) under the \(l\)-adic representation on the Tate module of \(A\) is connected. The field \(F'(A)\) has many interesting properties. For instance, \(F'(A)\subset F(A_n)\) for any \(n\geq 3\) and \(F(\text{End}(A)) \subset F'(A)\). The authors produce examples when \(F(\text{End} (A))\neq F'(A)\). Before that they give some description of the extension \(F'/F\) for abelian varieties \(B\) such that \(B\) is an \(F\)-form of an abelian variety \(A\) with \(F(\text{End}(A))=F'(A)\) and \(F(\text{End}(B))=F\).