A survey of special curves and special subvarieties of moduli space, unifying Hodge-theoretic, Teichmüller, and bi-algebraic perspectives, with a few new results and conjectures.
Abelian varieties are not quotients of low-dimension Jacobians
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abstract
We prove that for any two integers $g\geq 4$ and $g'\leq 2g-1$, there exist abelian varieties over $\overline{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g'$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.
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What makes an algebraic curve special?
A survey of special curves and special subvarieties of moduli space, unifying Hodge-theoretic, Teichmüller, and bi-algebraic perspectives, with a few new results and conjectures.