19 problems

Let $b<0$, and consider the minimization problem in $A_{T/2}$ with fixed mass $m$, denoted by … operatorname{sn}{alpha,beta,k}(x)=frac{1}{alpha}operatorname{sn}left(frac{x}{beta},k…

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…

Cohn polynomial conjecture. As a polynomial in $J$ with integer coefficients, no Cohn polynomial has a nontrivial factor, except for the Type I Cohn polynomials with…

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 $s$ and $n$ be nonnegative integers. Let $t_1,\dots,t_s$ be $\mathbb{Q}$-linearly independent alg…

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…

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 $\vec{c}=(c_1,\dots,c_n)=(1^L2^M3^N)$ 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…

Numerical evaluation. We believe that

For every positive integer $p$, let $i_p$, $j_p$, $g_p$, and $h_p$ be the numbers appearing in Theorems and. Coefficient relations. The following identities should hold: … The conj…

Let $H_N$ be the Fateev–Zamolodchikov spin-chain Hamiltonian, with twist angle $\phi$ and total third spin component $S_N^3$. The elliptic zero-mode conjecture. For odd $N=2n+1$, t…

Homothety conjecture. All infinity-shaped extremal curves are homothetic to the curve $\gamma_\alpha$ for this choice of $\alpha$, where $\xi\approx.90890856$.

Let $\Lambda\subset\mathbb{C}$ be a lattice, let $\lambda\in\Lambda\setminus\{0\}$, and let $\omega\in\mathbb{C}\setminus(\mathbb{Q}\lambda\cup\Lambda)$. Waldschmidt's conjecture.…

Uniqueness conjecture. For a symmetric monopole, the denominator of $Q_0(z)$ has $2(|n_1|-1)$ zeros, and consequently the tetrahedrally symmetric monopole is the only monopole in t…