The Existence of an Abelian Variety over the Algebraic Numbers isogenous to no Jacobian

We prove the existence of an Abelian variety $A$ of dimension $g$ over $\text{Qa}$ which is not isogenous to any Jacobian, subject to the necessary condition $g>3$. Recently, C.Chai and F.Oort gave such a proof assuming the Andr'e-Oort conjecture. We modify their proof by constructing a special sequence of CM points for which we can avoid any unproven hypotheses. We make use of various techniques from the recent work of Klingler-Yafaev et al.