For any genus, as long as you have non-ismorphic curves, you have non-isomorphic Abelian variety.
For genus 2, you have the Honda-Tate theorem, giving the classes of Abelian varieties (and there is a recent paper Howe, Maisner, NArt Nart and Ritzenthaler saying which of them are p.p.a.vs)
- Fro genus > 2, I think classification is at the forefront of current research.
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