14 problems

Let $g\geq 2$. The partial compactification ${\cal A}_g^{\ast}$ has Picard group generated modulo torsion by the ${\mathbb{Q}}$-line bundle $L$ of modular forms of weight $1$ and t…

Conjecture 5.3.5. For the case $g=3$, $\mathfrak i=2$, any $\sigma_1,\sigma_2\in\mathfrak D_1\cap S(3,2)^{open}$ satisfy 4.2.6.

Conjecture 1.6.3. The $D$-partition of $S(g,\mathfrak i)$ is a subpartition of the $\mathfrak D$-partition.

Conjecture 1.5.6. The restriction of the $I$-partition to $\goth D_{2g}$ is the intersection of the $\goth H$-partition with the restriction of the $D$-partition to $\goth D_{2g}$.

Conjecture 1.5.4. The $D$-partition of $S(g)$ is a subpartition of its $\goth D$-partition. Two elements $\sigma_i,\sigma_j\in\goth D_j$ are $D$-equivalent if and only if the quadr…

Let $1<r\leq n/2$, let $\mathcal Z_r(m)$ be the special cycles on $\mathcal A_n$ associated with principally polarized abelian varieties of dimension $r$ as isogeny factors, and de…

Virtual modularity conjecture. The series $\Theta^{2,\mathrm{vir}}_{\mathcal A_n}$ is a holomorphic Siegel modular form of weight $2n$ for $\operatorname{Sp}_4(\mathbb Z)$, valued…

Greer–Lian's modularity conjecture. The series $\Theta_{\mathcal A_n}(q)$ is a holomorphic modular form of weight $2n$ for $\operatorname{SL}_2(\mathbb Z)$, valued in…

The coherent cohomological dimension conjecture. It is conjectured that

Equivalent modularity criteria. The following statements are equivalent:

Let $g\geq 1$, let $d\in\mathbb{Z}_{\geq 0}$, and let $V$ be a Hodge generic irreducible subvariety of $\mathcal{A}_g$ of dimension less than $\dim(\mathcal{A}_g)-d$, defined over…

Let the $GSp_{4}$-system be the dynamical system under consideration, and let a $\textmd{KMS}_\beta$ state mean a KMS state at inverse temperature $\beta$. The two cases in questio…

Let $\Gamma\subset \operatorname{GSp}_2(\mathbb{Q})$ be an arithmetic subgroup, and set … Assume that $a\ge b\ge0$ and that $V_\Gamma$ has Hodge numbers satisfying … Suppose that…

Let $S$ be a Siegel threefold defined over $\mathbb{Q}$, let $\gamma$ be a Hecke correspondence on $S$ defined over $\mathbb{Q}$, and retain the notation for the coefficient system…