Local to global principles for homomorphisms of abelian schemes (Q6584659)

Let \(S\) be a smooth variety over a finitely generated field \(k\) of arbitrary characteristic. Let \(\mathcal A\) and \(\mathcal B\) be \(S\)-abelian schemes with generic fibers \(A\) and \(B\), respectively, defined over \(k(S)\). Let \(U\) be a dense open subscheme of \(S\), and let \(m\) be a non-negative integer \(\le \dim(S)\). The authors of the paper under review prove that if \(k\) is infinite or \(m>0\), then the following are equivalent: (1) There exists \(k(S)\)-isogeny \(A\to B\); (2) For every \(m\)-dimensional smooth connected subscheme \(T\) of \(U\) there exists a \(k(T)\)-isogeny \(A_T \to B_T\) where \(A_T\) and \(B_T\) denote the generic fibers of the base changed abelian schemes \(\mathcal A_T \to T\) and \(\mathcal B_T \to T\), respectively. They also prove similar statements for \(A\to B\) being surjective homomorphisms, non-zero homomorphisms, and homomorphisms with \(\kappa\)-dimensional kernel where \(\kappa>0\).\N\NTheir second main result is a similar statement for quadratic isogeny twists. For a field \(K\), an abelian variety \(B/K\) is called a quadratic isogeny twist of an abelian variety \(A/K\) if there is a quadratic twist \(A'/K\) of \(A\) and a \(K\)-isogeny \(B\to A'\). The authors prove that if \(k\) is infinite or \(m>0\), then the following are equivalent: (1) \(A\) is a quadratic isogeny twist of \(B\); (2) For every \(m\)-dimensional smooth connected subscheme \(T\) of \(U\) the abelian variety \(A_T\) is a quadratic isogeny twist of \(B_T\).\N\NFor the case where \(k\) is finite and \(m=0\), they prove a similar result about the quadratic isogeny twists under the assumption that both \(A\) and \(B\) have the trivial \(\overline{k(S)}\)-endomorphism rings, and also prove a result about \(\overline{k(S)}\)-isogenies and homomorphisms under the assumption that Zarhin's Miniscule Weights Conjecture is true.