7 problems

Let $\mathcal{A}_g$ be the moduli space of principally polarized complex abelian varieties of dimension $g$, let $T_g$ be the Torelli locus, and let $T_g^o$ be its open Torelli loc…

Let $g$ be a sufficiently large genus, and let a smooth projective curve of genus $g$ have a Jacobian of ST type, as in the source. Coleman's conjecture. Only finitely many smooth…

Low-genus density conjecture. If $g\leq 6$, then for each $K$, $E_{g,K}$ is dense in $\mathcal{M}_g(\mathbb{C})$. This is the complementary low-genus prediction to the high-genus f…

Real multiplication finiteness conjecture. Fix $g\geq 7$. Each $E_{g,K}$ is contained in a finite union of Hilbert modular varieties. Moreover, $E_{g,K}$ is empty for all but finit…

Let $g\geqslant4$ be an integer, let $S\subset\mathcal A_g$ be a closed Hodge generic subvariety of dimension $q$, and let $\mathbb V$ be the induced polarizable $\mathbb Z$-VHS on…

For $i=1,2$, let $g_i\in\mathbb{Z}_{\ge 1}$, and let $\nu_i$ be a symmetric Newton polygon appearing on the open Torelli locus $\mathcal{T}_{g_i}^{\circ}$. Write … Let $\nu$ be the…

Let $M\subset\mathcal{A}_g$ be a Shimura subvariety of the unitary type specified earlier in the source. Unitary disjointness conjecture. The intersection … is empty. The conjectur…