22 problems

Eisenstein conjecture. Each Weyl group multiple Dirichlet series appears as the first Fourier-Whittaker coefficient in an Eisenstein series on the metaplectic $n$-fold cover of the…

Gelfand–Tsetlin p-part conjecture. The $p$-part constructed from $f(\mathbf{x};\ell)$ should coincide with the $p$-part arising from the Gelfand–Tsetlin-pattern construction.

Let $A$, $\boldsymbol{\Pi}$, $\boldsymbol{\omega}$, and $\boldsymbol{\omega'}$ be parameters for the Dirichlet series…

Let $A$ be the integer matrix defining the multiple Dirichlet series $D_{A,\boldsymbol{\Pi}}(\boldsymbol{s};\boldsymbol{\omega},\boldsymbol{\omega'})$, let…

Let $V$ be a variety, and let Property (S) denote the property introduced in the paper that permits construction of multiple Dirichlet series with Euler product expansions by restr…

Consider a multiple Dirichlet series obtained by restricting a summation to a variety, with an Euler product whose local factors cannot be generated by finitely many terms. Discret…

Elliptic-curve multiple-Dirichlet-series pole conjecture. The function

Unitary multiple-Dirichlet-series pole conjecture. The function

CFKRS conjecture. If $|\operatorname{Re}(s)-\frac12|\ll1/\log Q$ and $|\operatorname{Re}(z)-\frac12|\ll1/\log Q$ for all $s\in S$ and $z\in Z$, then as $Q\to\infty$,

Eisenstein conjecture. The Whittaker coefficients of a metaplectic Eisenstein series are Weyl group multiple Dirichlet series; in the setting above, the first Whittaker coefficient…

Absolute-convergence conjecture. The series

Holomorphy conjecture. There exists $\mathbf I(\mathfrak s,t)\in\mathscr S_W$, symmetric in $\mathfrak s_1,\ldots,\mathfrak s_r$ and nonvanishing on $X_0^*$ for every $t>1$, such t…

Absolute-convergence conjecture. The series defining the decomposition is absolutely convergent in this region away from the poles of $Z(s_1,\ldots,s_{r+1},1;h)$.

Meromorphic-continuation conjecture. These twisted Weyl-group multiple Dirichlet series admit meromorphic continuation to the interior of $X_0^*$ and are holomorphic there except f…

A multiple Dirichlet series of type $A_r$ is associated with the corresponding Weyl-group functional equations and metaplectic automorphic structure. Type $A_r$ correspondence conj…

Diagonal residue product conjecture. We have

Weyl group multiple Dirichlet series are series in several complex variables associated with root systems, whose functional equations form a group isomorphic to the Weyl group. Fou…

Let $H_{\mathfrak{c}}$ be the ray class group modulo $\mathfrak{c}$, let $R_{\mathfrak{c}}:= H_{\mathfrak{c}} \otimes \mathbb{Z}/n\mathbb{Z}$, and let $\psi$ and $\psi'$ be charact…

Odd-degree conjecture. The integer $n$ should be odd.

Type C multiple-Dirichlet-series conjecture. The series $Z_\Psi(\mathbf{s};\mathbf{m})$ should have meromorphic continuation to $\mathbb{C}^r$ and satisfy a group of functional equ…

Metaplectic Eisenstein-series conjecture. The families of multiple Dirichlet series associated to $\Phi$ should be precisely the Fourier–Whittaker coefficients of minimal parabolic…

Let $n$ be a general positive integer, and let the degree-$n$ $A_n$ Weyl group multiple Dirichlet series be given. Let the series constructed by Friedberg, Hoffstein, and Lieman fr…