131 problems

Belfiore–Solé conjecture. For every unimodular lattice $L$, the function $\Xi_L(y)$ attains its global maximum on the positive imaginary axis at the symmetry point $y=1$.

Merca's conjecture. This theta series has non-negative coefficients. The conjecture refines the established positivity theorem for averaged truncations of Jacobi's triple product i…

Frobenius structure conjecture. The coefficient of $\vartheta_0$ in the product

Let $p_8$ denote the generalized octagonal-number polynomial, and let $b,c,d$ be chosen from the five triples … A sum $p_8+bp_8+cp_8+dp_8$ is called universal over $\mathbb{Z}$ whe…

Stronger linear lower-bound conjecture. For each prime $k>3119$,

Liouville's conjecture. The number of representations satisfies

Let the Jacobi theta function of a three-dimensional lattice be the theta function referred to in the source, and let FCC and BCC denote the face-centered-cubic and body-centered-c…

Let $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$. Let $H^{q,k}(z,v,\omega,\lambda)$ be the integral defined by … where $J$ and $\mathfrak{g}$ are…

Let $Th_-$ be the map from the moduli space of triples $(A,L,\vartheta)\in\mathcal{A}_2(3)^-$ to $\mathbb{P}^3_-$ introduced above. Its four components are functions on the releva…

Let $q=2^d$, let $\mathbb{Z}_q$ be the Witt vectors with values in the finite field $\mathbb{F}_q$, and let $\sigma$ be the canonical lift of the absolute $2$-Frobenius. Let $E$ be…

Mestre's AGM sequence is a sequence of theta-null points associated with Mestre's arithmetic–geometric mean construction. A moduli interpretation conjecture. Mestre's AGM sequence…

For a principally polarized abelian variety, let $Th_2$ denote the theta map associated with second-order theta functions. Injectivity conjecture for $Th_2$. The map $Th_2$ is inje…

Let $X$ be a hyperelliptic Riemann surface of genus $g\geq 2$, let $\tau$ be a period matrix associated to $X$, and let … be a fundamental system, where $\mathcal{F}$ is the set of…

Let $\mathbb R_N$ and $\mathbb R_{N-1}$ denote the parameter ranges defined in the paper, with $\mathbb R_N\subseteq\mathbb R_{N-1}$, and let the metric be Sklyanin's metric on the…

Let $\Theta_N$ be the space of theta functions equipped with the metric specified in the paper, and let $K_v$ denote its reproducing kernel at $v$. Sklyanin's reproducing-kernel co…

A multiple hypergeometric series is called totally elliptic when the ratios of its successive coefficients are elliptic in the summation variables and parameters. Modular invarianc…

Let $X$ be a smooth projective curve of genus $g$, let $SU_X(r)$ denote the moduli space of semistable rank-$r$ vector bundles with trivial determinant, and let $U_X(k,0)$ denote t…

The Baker–Akhiezer function is defined on a Riemann surface by prescribed meromorphicity and asymptotic properties, generalizing the common eigenfunctions of two commuting ordinary…

Let $\mathfrak g$ be one of the Lie algebras covered by Theorems 1 and 2, and let $\iota_{\mathfrak h}$ be the map whose image is contained in the spaces of holomorphic functions d…

Vanishing conjecture. The modular form $\Psi_4(\Omega)^2-2^h\Psi_8(\Omega)$ vanishes for every genus $h$.

Affine theta-function expansion conjecture. The $A_{N-1}$ theta function in the blow-up formula satisfies

Prym theta-function isomorphism conjecture. These homomorphisms should be isomorphisms in both the odd- and even-rank cases.

Positivity conjecture. For all integers $k\ge1$, $d\ge2$, and $n\ge0$, the Laurent polynomial $\mathscr{L}_{k,n}^{d}(z)$ has positive coefficients. This conjecture is presented as…

Level-rank duality conjecture. This map is an isomorphism. The source states that this conjecture was proven by Marian and Oprea and by Belkale, so it is recorded as solved.

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…