11 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 $K$ be a real quadratic field, let $p$ be a finite prime that is ramified or inert in $K$, and let $\tau$ be a real multiplication point in the Drinfeld $p$-adic upper half pla…

Uniformity conjecture for singular $S$-units. For every $s \in \mathbb{N}$, the set $\mathcal{J}_s$ is finite.

Finiteness conjecture for singular S-units. For every $s \in \mathbb{N}$, the number of singular moduli that are $S$-units for some set of rational primes $S$ with $\#S=s$ is finit…

The $p$-adic singular-moduli approximation conjecture. Every algebraic number is badly approximable in $\mathbb{C}_p$ by the singular moduli of the $j$-invariant.

Riffaut's conjecture. A singular modulus of degree $h\ge 3$ cannot be a root of a trinomial with rational coefficients.

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…

Powers-of-singular-moduli linear equation conjecture. Then one of the following holds: $A+B=C=0$, $x=y$, and $m=n$; $x,y\in\mathbb Q$; or $x\ne y$ and $\mathbb Q(x)=\mathbb Q(y)$ i…

Let $\chi^*$ be the level $1$ almost holomorphic modular function defined by … where $E_4$ and $E_6$ are the usual Eisenstein series and…

Let $d_1$ and $d_2$ be quadratic imaginary discriminants with relatively prime conductors. Write $f$ for the product of the two conductors, and for each prime $p$ let…