9 problems

Let $\mathcal{M}_g$ be the moduli space of smooth projective curves of genus $g$ over $\mathbb{C}$, let $\mathcal{A}_g$ be the moduli space of $g$-dimensional principally polarized…

Let $g',g”\in\mathbb Z_{>0}$, and let $\xi'$ and $\xi”$ be symmetric Newton polygons appearing on $\mathcal T^0_{g'}\otimes\mathbb F_p$ and $\mathcal T^0_{g”}\otimes\mathbb F_p$, r…

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…

Let $g$ be sufficiently large, let $M$ be a weakly special subvariety of the moduli space $44\mathcal{A}_g44$, and let $H(M)$ denote its total Hecke orbit. Finite Hecke-translate c…

The corrected Coleman conjecture. For $g\geq 8$ there are only finitely many smooth projective curves $C$ over $\mathbb{C}$ of genus $g$ such that $J(C)$ is an abelian variety of C…