41 problems

Let $A$ be a configuration, let $f$ be a rational $A$-hypergeometric function in variables $x_1,\dots,x_s$, and let a facial subset be a subset of $A$ arising as a face of the conf…

Let $\Delta_k$ denote the $k$-simplex, and let … where $k_0\geq\cdots\geq k_r$. The configuration $A$ is GKZ-rational when its associated GKZ hypergeometric system has the rational…

Let $A_0,A_1,\dots,A_r$ be vector configurations in $\mathbb{Z}^r$, and let $A$ be their Cayley configuration … where $e_0,\dots,e_r$ is the standard basis of $\mathbb{Z}^{r+1}$. T…

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and define … where $F(a,b;c;z)$ is the hypergeometric function and $k$ is complex. The zero-free region conjecture. For ever…

Let $n$ be the dimension parameter, let $\rho=(n-1)/2$, and let $F(a,b;c;z)$ denote the hypergeometric function. For the function $F_k(z)=F(k,-\rho;k+n/2;z)$, with complex variable…

Generalized hypergeometric expression conjecture. For a given $\ell\geq 0$, the components of the functions $H(t)$ are expressible in terms of generalized hypergeometric functions…

For a positive integer $m$, let $\widetilde X_{12}^{(a)},\widetilde X_{13}^{(a)},\widetilde X_{23}^{(a)}$ for $1\leq a\leq m$ be the formal variables associated with the multiple g…

Wang's transformation conjecture. If $p$ is an odd prime, then

Let $\mathcal{K}(r)=(\pi/2)F(1/2,1/2;1;r^2)$ be the complete elliptic integral of the first kind. For $r\in(0,1)$, set $r'=\sqrt{1-r^2}$. Anderson–Vamanamurthy–Vuorinen's conjectur…

Let $a,b>0$ and $r\in(0,1)$. Let $\Lambda(a,r)$ and $\Lambda_2(a,b_1,b_2,r)$ denote the quantities defined earlier in the source by the inequalities referenced as … , respectively.…

Let $F(a,b;c;r)={}_2F_1(a,b;c;r)$ denote the Gaussian hypergeometric function, defined for the parameters under consideration by … For $a>0$, $b>0$, and $r\in(0,1)$, consider the r…

Special-value conjecture.

For positive integers $d$ and $N$, and $\lambda \in L(\mathbb{C})$, consider the smooth hypersurface $U_{N,\lambda}^{(d)} \subset \mathbb{A}^{d+1}$ defined by … The group…

For every prime $p\ne3$, consider the truncated hypergeometric sum … Let $a_p$ be the coefficient of $q^p$ in the weight $4$ cusp form of level $54$ beginning…

Generic-parameter extension conjecture. The formula holds for generic values of $a,c\in\mathbb{C}$ and all nonzero $m\in\mathbb{Z}$.

Let $p$ and $q$ be integers with $p<q$, let ${\mathbf b}=(b_1,\ldots,b_q)$ satisfy $b_i>0$, and let ${\mathbf a}=(a_1,\ldots,a_p)$ satisfy $a_i\geq b_i$ for $1\leq i\leq p$. The ge…

Let $(r_1,r_2,q)$ range over … Set $\mathbf{\alpha}_{(r_1,r_2)}=\{r_1,1-r_1,r_2,1-r_2\}$ and $\mathbf{\beta}_q=\{1,1,q,2-q\}$. Let $\mathrm{HD}_{(r_1,r_2,q)}$ denote the hypergeome…

Elliptic-family p-adic L-function conjecture. There is a constant $C_{n,a}\in {\mathbb Q}^{\times}$ independent of $p$ such that

The $a=1$ p-adic L-function conjecture. There is a constant $C_1\in {\mathbb Q}^{\times}$ independent of $p$ such that

Irreducibility conjecture. The polynomial $R_n(x;\ell_1,\ell_2,y)$ is irreducible unless $\ell_1=\ell_2$ and $n$ is odd. In that exceptional case, $R_n(x;\ell_1,\ell_2,y)$ is the p…

Let $\Omega_h$ be a polygon whose boundary consists of edges of a hexagonal lattice of meshsize $h$, with distinct side midpoints $a_h,b_h$ and lattice-edge midpoints $u_h,v_h$. Co…

Let $\alpha\in{\mathbb Q}\setminus\{0,1\}$ and let $X_\alpha$ be the elliptic curve in the preceding construction. Let $\xi_{\mathrm{Ross}}\in K_2(X_\alpha)^{(2)}$ be the higher Ro…

Let $s\geq 1$ be an integer and let $a\in\mathbb{Z}_p$. Dwork's $p$-adic hypergeometric function is … where the numerator and denominator are hypergeometric power series. Suppose…

Let $W$ be the coefficient ring, let $\sigma(t)=ct^p$ and $\widehat{\sigma}(t)=c^{-1}t^p$, and let $a\in\mathbb{Z}_p\backslash\mathbb{Z}_{\leq 0}$ satisfy $a^{(r)}=a$ for some…

Convexity conjecture. For every $a,b>0$, the set ${\mathcal{D}}_{a,b}$ is convex.