13 problems

Let $\tau,\omega\in\mathcal{H}_p$ be RM points, with $\tau\notin\Gamma\cdot\omega$, and let $J_p(\tau,\omega)\in\mathbb{C}_p^\times$ be the value of the associated rigid meromorphi…

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…

Let $J_{E}^{+}[\tau]$ denote the construction in the source applied to RM points, and let a ring class field be the corresponding class field. Caiero–Darmon Conjecture 7. The image…

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

Fix $g$ and let $\eta\in\overline{\mathbb Q}$ be such that $\mathbb Q(\eta)$ is a totally real number field of degree $g$. For sufficiently large prime powers $q=p^n$ with…

Let $c>0$ be the constant in the expected time and space complexity bound for the probabilistic Las Vegas algorithm that computes the local zeta function of a genus-$g$ hyperellipt…

Let $V_a^{(2)}$, $V_a^{(5)}$, $V^{(13)}$, $V_{a,b}^{(2)}$, $V_{a,b}^{(3)}$, $V_{{\bf a},{\bf b}}^{(-1)}$, $V^{(-1,\mu_7)}$, $V^{(-1,\mu_9)}$, and $V^{(-1,\mu_{19})}$ be the $K3$ su…

Let $A$ be an abelian variety over $\mathbf{Q}$ with an endomorphism ring that embeds into a totally real field. For an integer $d\geq 1$, and a prime $p$, let $K$ be a finite exte…

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…