10 problems

Let $N$ be the level parameter, let $\alpha$ be the invariant, let $\operatorname{image}\alpha$ denote its image, let $\Phi_m^{\alpha}$ be the modular polynomial with level structu…

Let $N$ be the level parameter, let $\alpha$ be the invariant defining the modular polynomials with level structures, and let $J_1$ be the corresponding polynomial. For a polynomia…

Let $ell$ be a prime with $0<m<ell$, and let $a_{ell,ell-m}$ be the coefficient of the $ell$-th classical modular polynomial $Phi_ell(X,Y)$. Suppose one of the following holds:…

Congruence conjecture. The following divisibilities hold: if $ell\ne 2$, then $a_{m,n}\equiv 0\pmod{2^{15c}}$; if $ell\ne 3$, then $a_{m,n}\equiv 0\pmod{3^{3c}}$, and if additional…

Russell's conjecture for level 2. There exists a polynomial $P(u,v)$ of degree $(p+1)/2$ in $u$ and $v$ such that

Resultant multiplicity conjecture. The following congruences hold modulo $p$:

Denominator conjecture. The polynomial $L'_p$ is the denominator of every coefficient of the three modular polynomials in the functions $b'_1,b'_2,b'_3$.

Heuristic height-bound conjecture. The bound

Asymptotic height lower-bound conjecture. For prime $l$,

Height upper-bound conjecture. For every prime $l>30$,