39 problems

Let a prime-detecting quasimodular form be a quasimodular form whose Fourier coefficients detect primes. A quasimodular Eisenstein series is a quasimodular form in the Eisenstein s…

For even weight $w$ and depth $s$ with $0\leq s\leq w/2$, $s\neq w/2-1$, let $X_{w,s}=\sum_{n\geq 0}a_nq^n$ be a normalized extremal quasimodular form of weight $w$ and depth $s$,…

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 $F_{\leq k}\mathcal{M}_{\mathbb{Z}}:=F_{\leq k}\mathcal{M}\cap\mathbb{Z}[[q]]$ 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 $\lambda$ be a straight partition shape and let $\mu\subseteq\lambda$ be a subdiagram. In the transition expansion of a Schur Eisenstein series, write $d_\lambda(\mu)$ for the…

Let $g(\lambda)$ denote the Schur MacMahon series indexed by partitions $\lambda$, and let $\mathcal{M}_{\mathbb{Z}}$ be the quasimodular forms with integral Fourier coefficients.…

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…

Let $F = \sum_{n \ge n_0} a_n q^n$ be a positive quasimodular form, and let $\{n: a_n > 0\}$ be the set of indices with positive Fourier coefficient. Its natural density is the lim…

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 … Here $U_{2n}(q)$ is the $q$-series defined from Jacobi's weight $3/2$ theta function. Ramanujan's quasimodularity conjecture. Each $U_{2n}(q)$ should be a weight $2n$ quasimo…

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 $\pi:X\to B$ be a Calabi–Yau fibration with relative dimension one and a section, and let $F_{g,\alpha}$ denote its $\pi$-relative Gromov–Witten potentials. Elliptic-fibration…

The Gromov–Witten potentials involve the generating functions of Gromov–Witten invariants of Enriques and bielliptic surfaces. Quasimodularity conjecture. The Gromov–Witten potenti…

Ramanujan's conjecture. Every $U_{2k}$ has an expression in terms of the renormalized Eisenstein series $E_{2j}$. Ramanujan's formulas anticipate quasimodular expressions for these…

Let $f_w$ be the functions in the Feigenbaum–Grabner–Hardin family, and call a form completely positive when its relevant Fourier expansions at the cusps have positive coefficients…

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…

Let $\mathcal{T}$ be a four-dimensional $\mathcal{N}=2$ superconformal field theory, let $\mathcal{I}_{\mathcal{T}}(q)$ be its Schur index, and let $\mathbb{V}(\mathcal{T})$ be the…

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…

Let $G_w^{(r)}$ be the normalized extremal quasimodular form of weight $w$ and depth $r$, and let $\theta_k^{(r)}$ denote the differential operator used in the source. Kaneko and K…

Kaneko–Koike's conjecture. Extremal quasimodular forms should exist, and should be unique when normalized, for pairs $(w,r)$ satisfying the naturally required numerical constraints…