10 problems

Fix $d\in\mathbb Z_{>0}$, $d\neq3$, write $\Delta=\Delta_d=(d+1)(d-3)=f^2\Delta_0$ with $\Delta_0$ a fundamental discriminant, and let $\mathcal M(f')$ be the multiplet indexed by…

Let $u\in S_d$, write $\nu=u^{1/3}$, and let $c=c(u)$ be the product of the distinct rational primes dividing $u$ but not $d$. Let $M_{3c}$ denote the ring class field associated w…

Let $u\in S_d$, write $\nu=u^{1/3}$, and let $c=c(u)$ be the product of the distinct rational primes dividing $u$ but not $d$. Let $c'$ be the odd part of $c$. Let…

Let $d$ be as in the paper, let $\mathcal{O}(d)$ be the associated quadratic order, and let $P_d$ and $R_d$ be the subsets defined there. For $\beta\in\mathcal{O}(d)$, let…

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_i/F$ be quadratic extensions in which $\mathfrak p$ is inert, wit…

Let $p$ be a prime number, let $v_p:4^*\to\mathbb{Z}$ be the usual $p$-adic valuation, and let $D=-3f^2$ be a discriminant. Let $j_D$ be any singular modulus relative to $D$, and l…

Let $l>3$ be a prime with $l\equiv 3\pmod{4}$, let ${\mathsf{k}}=\mathbb{Q}(\sqrt{-l})$, and let $p$ be a prime that is non-split in ${\mathsf{k}}$. Let $\mathfrak{P}_0\mid p$ be t…

Let $\uptheta\in\mathbb R-\mathbb Q$ be a fundamental quadratic unit and let $K=\mathbb Q(\uptheta)$, with fundamental discriminant $D$. The multivalued quantum modular invariant…

Darmon's conjecture. The local point $P_\tau=\Phi_{\operatorname{Tate}}(J_\tau)$ belongs to $E(H_\tau)$, where $H_\tau$ is the ring class field associated with the order determined…

Let $Y$ be an Enriques surface whose universal cover $X$ is a singular K3 surface. Let $d<0$ denote the discriminant of $X$, and let $H(d)$ and $H(4d)$ denote the corresponding rin…