62 problems

Let $\Omega$ be a lattice in $\mathbb{C}$, let $k$ be the associated elliptic curve's endomorphism field, and let $g_2,g_3$ be its invariants. Let $p_1,\dots,p_n$ be $k$-linearly i…

The Jacobian elliptic functions $sn$ and $cn$ are combinatorial generating functions whose coefficients involve both factorial and Euler numbers. Schützenberger's conjecture. Jacob…

Elliptic Macdonald polynomials are the quotients of hypergeometric theta functions by suitable products of ordinary theta functions, arising in the q-deformation of conformal block…

Generalized Lamé operators conjecture. The generalized Lamé operators are all algebraically integrable.

Correspondence conjecture. The ultradiscretized elliptic functions can be realized as particular solutions of the tropical elliptic functions.

Let $s$ be real and range over a bounded interval, and let $\mathcal{B}_{2k+1}(s)$ denote the elliptic Bernoulli polynomial of degree $2k+1$, with derivative…

Let $\mathcal{F}_m(\lambda)$ denote the elliptic Faulhaber polynomial and let $\mathcal{B}_{2m-2}$ denote the corresponding elliptic Bernoulli number. For real $x$ in a finite inte…

Let $\mathcal{B}_{2m}$ be the elliptic Bernoulli numbers, and let $w$, $\eta$, $g_2$, and $g_3$ denote the associated Weierstrass quantities. Positivity conjecture. The elliptic Be…

Let the deformed quantum Calogero–Moser operators be generalized by replacing the trigonometric interaction function $1/\sin^2 z$ with the Weierstrass elliptic function $\wp(z)$. T…

Elliptic Cherednik–Macdonald–Mehta conjecture.

Let $\Omega$ be a lattice in $\mathbb{C}$. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent complex numbers, and let $q_1,\dots,q_r,p_1,\dots,p_n$ be $k$-linearly independe…

Let $E_{\tau}$ be the rectangular torus associated with a fixed parameter $\tau$, and let $G_p(z)$ denote the Green function with parameter $p\in E_{\tau}$. Conjecture A. Fix any…

Let $G_p(z;\tau)$ be the Green function considered in the paper, with $p$ and $\tau$ as in the preceding results. Conjecture A. If $G_p(z;\tau)$ has $10$ critical points, then thes…

The Kaup–Newell hierarchy contains the derivative nonlinear Schrödinger equation and related integrable systems. An $N$-elliptic localized solution is represented in the form estab…

Let $r$ be a positive integer, let $\alpha$ be an $r$-core, and let $a_\lambda(\square)$, $l_\lambda(\square)$, and $h_\lambda(\square)$ denote the arm length, leg length, and hook…

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $p\in\mathbb{C}\smallsetminus\Omega$ with algebraic $\zeta(p)$, and let $\alpha$ be an algebraic…

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $s,m,n$ be nonnegative integers with $0\leq m\leq n$. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearl…

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants $g_2,g_3$. Let $s$ and $n$ be nonnegative integers. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent alg…

Split semi-elliptic conjecture. The transcendence degree of $K$ is at least $s+2n$, unless $2\pi\mathrm{i}\mathbb{Q}\subseteq\mathbb{Q}t_1+\cdots+\mathbb{Q}t_s$ and…

Consider elliptic cluster algebras associated with elliptic generalizations of cluster mutations, and elliptic isomonodromic deformation flows. Elliptic-mutation conjecture. Ellipt…

Let $q$-isomonodromic systems have monodromy manifolds, and let elliptic dilogarithms be the generating functions associated with the proposed transformations. Elliptic cluster str…

Let $N$ be a positive integer, and let $x_i$, $p_i$, $m$, $\beta$, $\mathfrak{q}$, and $p_{6d}$ be the parameters and variables appearing in the generating function … For…

Let $\vec{c}=(c_1,\dots,c_n)=(1^L2^M3^N)$ and suppose that $q_{\vec{c}}\neq1$. Let $f^{\vec{c}}_{11}$ be the corresponding elliptic corner-VOA structure function and…

Let $\Gamma=\Gamma(1/4)$ and, for $n\geq1$, define … Here $\mathbb{Q}$ denotes the rational numbers. Fifth-moment arithmeticity conjecture. For every integer $n\geq1$, … and … Thes…

Let $\Gamma=\Gamma(1/4)$. The Berndt-type integral … should equal … This is a numerically verified special case of the paper's broader arithmetic-pattern results, but no proof of t…