19 problems

Let $F_{\leq k}\mathcal{M}_{\mathbb{Z}}:=F_{\leq k}\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral quasimodular forms of filtered weight at most $k$, and let $g(\lambda)$ be the Sc…

Let $\nu\subseteq\lambda$ be partitions, let $d_\lambda(\nu)$ be the corresponding transition coefficient, and let $F_{\lambda\nu}$ be the associated element of the symmetric-funct…

Let $\mathcal{M}_{\mathbb{Z}}:=\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral lattice of quasimodular forms, and let $\mathcal{G}_{\mathbb{Z}}:=\sum_\lambda\mathbb{Z}\,g(\lambda)$…

Let $X_{w,2}(z)$ be the depth-two extremal quasimodular form of weight $w$, and define … A quasimodular form is completely positive when all coefficients in the relevant complete-p…

Let $w$ be an even integer with $w \ge 6$ and $w \ne 8,10$, and let $X_{w,1}$ denote the extremal quasimodular form of weight $w$ and depth $1$. Monotonicity conjecture. For all ev…

Optimal Sturm bound conjecture. The form $f$ is determined uniquely by its first $\dim \widetilde{M}_{2k}$ Fourier coefficients. Furthermore, for every integer $m\geq 2$, the reduc…

Let $N$ be a positive integer. Let $\beta\widetilde{E}(N)$ denote the space of quasimodular Eisenstein forms on $\Gamma_{0}(N)$, let $\Omega_{N}$ denote the space of prime-detectin…

Let $M=U\oplus U(2)\oplus E_8(-2)$ be the Enriques lattice, let $\mathcal D$ be the relevant orthogonal domain, and let $\mathcal H_{-2}\subset\mathcal D$ be the norm $-2$ Noether–…

Let $\mathcal{E}$ be the quasimodular Eisenstein space, generated additively by the even-weight Eisenstein series and their derivatives. Let $\Omega$ be the space of prime-detectin…

Uniform-minor conjecture. For every $k\geq 14$, there exists an integer $n>\operatorname{dim}(\operatorname{QM}_k)$ such that $\mathcal{M}_k\mid S^k$ has…

Full-rank conjecture. The rank of $\mathcal{M}_k$ is always equal to $\operatorname{dim}(\operatorname{QM}_k)$. Equivalently,

Let $p\ge5$ be prime, and write … where $m\in\mathbb{Z}_{\ge0}$ and $\delta,\varepsilon\in\{0,1\}$. Let $A_{u,a}^{(r)}(X)$ be the generalized Atkin polynomial associated with the n…

Fix an integer $g\ge2$. Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and for each $d\ge1$ let $\phi_{g/1,d}$ be the morphism from the compactif…

Kaneko–Koike's conjecture. If $\ell\leq 4$ and $w>2$, then every Fourier coefficient $a_n$ is positive. Moreover, no prime factor of the denominator of any such coefficient is grea…

Higher-genus quasimodularity conjecture. The statement of this theorem holds for all $g\ge2$.

Coefficient-positivity conjecture. Every coefficient in this expansion is positive:

Quasimap elliptic-fibration conjecture. For $\gamma\in H_2(B,\mathbb Z)$,

Oberdieck–Pixton conjecture. For $\gamma\in H_2(B,\mathbb Z)$,

Let $X$ and $B$ be nonsingular projective varieties, let $cpi:X\to B$ be an elliptic fibration with integral fibers and a section $ciota:B\to X$, and let $N_{\iota}$ be the normal…