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

TL;DR

This paper proves the existence of certain Abelian varieties over algebraic numbers that are not isogenous to any Jacobian, using a new approach that avoids unproven hypotheses and builds on recent techniques.

Contribution

It provides a constructive proof of the existence of such Abelian varieties without relying on the André-Oort conjecture, advancing understanding in algebraic geometry.

Findings

Existence of Abelian varieties over algebraic numbers not isogenous to Jacobians for g > 3

Construction of special CM points avoiding unproven hypotheses

Application of recent techniques from Klingler-Yafaev et al.

Abstract

We prove the existence of an Abelian variety $A$ of dimension $g$ over $\Qa$ 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.