7 problems

Let $\tau,\omega\in\mathcal{H}_p$ be RM points with $\omega$ not in the $G$-orbit of $\tau$, let $\mathrm{T}\in\mathcal{I}_p$ be a Hecke operator, and let…

Poonen–Rains conjecture for real multiplication. If $p$ is odd, then, as $(A,\iota,\lambda)$ varies in $\mathrm{RM}_{\mathcal {O}}(F)$ ordered by conductors,

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…

Uniqueness conjecture. For every ample $\mathbb Q$-line bundle $L$ on $X$, there exists at most one irreducible curve $C$ that is submaximal for $L$ if and only if either

Let $f$ be a weight $2$ newform for the paramodular group $K(N)$, not in the span of the Gritsenko lifts. Let $k_f$ be the totally real number field generated by the Hecke eigenval…

A Teichmüller curve in ${\mathcal M}_{3}$ is algebraically primitive when its trace field has degree $3$. Finiteness conjecture. There are only finitely many algebraically primitiv…