175 problems

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

Let $X$ be a scheme of finite type over $\mathbb{Z}$, let $d=\dim X$, and assume that higher Chow groups $CH^r(X,i)$ are defined for such schemes, have finite ranks, and satisfy…

The Frobenius matrix $F(\Gamma)$ is the matrix arising from the Frobenius action in the recursive computation of the zeta function, with parameter $\Gamma$. At infinity, a function…

Let $S\subseteq\mathbb{P}^d$ be a possibly singular del Pezzo surface, let $U\subseteq S$ be obtained by deleting exceptional divisors, and let $\rho_S$ be the relevant Picard rank…

Lichtenbaum–Milne conjecture. For every relevant integer $r$, as $t\to q^{-r}$,

Let $t=\varphi(s)$ be a meromorphic function on $\mathbb{C}$, and let $\zeta_{q,r}(s,t,z;\boldsymbol{\omega})$ and $\zeta_r(s,z;\boldsymbol{\omega})$ denote the corresponding $q$-a…

Let $X$ and $Y$ be schemes of finite type over $\mathbb{Z}$ whose generic fibres $X\otimes\mathbb{Q}$ and $Y\otimes\mathbb{Q}$ form a usual weak mirror pair of $d$-dimensional Cala…

Canonical-model zeta conjecture. If $M_{K_1}$ and $M_{K_2}$ are isospectral, then the Hasse–Weil zeta functions of the canonical models of $M_{tilde{K}_1}$ and $M_{tilde{K}_2}$ are…

Let $f:X\to\bb A^1_{\bb C}$ be as in the construction of the twisted topological zeta function, let $e\geq 1$, and write $Z^{(e)}_{{\rm top}}(s)$ for that rational function. A mono…

Let $X$ be a Calabi–Yau threefold with $h^{1,1}(X)\neq h^{1,1}_{\mathrm{toric}}(X)$, and set … Let $\chi$ be a character depending on the defining equation of a singular locus. Tor…

Let $R_{\mathbf{v}}(t)$ be a contribution to the zeta function, and let the degree degeneration measure the decrease in $\operatorname{deg}(R_{\mathbf{v}})$ caused by a singularity…

Let $R_{\mathbf{v}}(t)$ be a contribution to the zeta function associated with a monomial class $\mathbf{v}$, and let $\operatorname{deg}(R_{\mathbf{v}})$ denote its rational-funct…

Singularity-degeneration conjecture. The degeneration of $\operatorname{deg}(R_{\mathbf{v}})$ for each piece of the zeta function of a Calabi–Yau manifold at a singular point in mo…

Let a family of Calabi–Yau hypersurfaces have Hodge numbers $h^{2,1}$ and $h^{2,1}_{\mathrm{poly}}$, and let $R_{\mathrm{excep}}(t)$ denote a factor of the numerator of its zeta fu…

Let $R$ be a Gorenstein order over $\mathbb{Z}$. Let $R^\vee=\operatorname{Hom}(R,\mathbb{Z})$, and define Yun's Dedekind zeta function by … Here $L(s,R)$ is the cubic-order $L$-fu…

Let $s$ satisfy $\operatorname{Re}(s)>1$, and let $(k_1,\ldots,k_p)$ and $(l_1,\ldots,l_q)$ be distinct finite sequences of positive integers. Write $H_s^\star(k_1,\ldots,k_p)$ for…

Let $R:=A$, $B:=B_A$, and $H:=H_A$. For a positive degree $d$, let $\mathcal{I}(d)$ be the set of ideals in $B$ whose norm has degree $d$, and define the analytic function…

Strong Monodromy Conjecture. 1. If $f\in\mathbb{Q}[x_1,\ldots,x_n]$, then, for all but finitely many primes $p$, every pole $s_0$ of $Z_{f,p}(s)$ has $\operatorname{Re}(s_0)$ as a…

Let $\mathfrak{O}$ be the coefficient ring, let $A$ be an $n$-dimensional nilpotent $\mathfrak{O}$-algebra, and write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and id…

Let $\mathfrak{O}$ be the coefficient ring and let $A$ be a finite-dimensional $\mathfrak{O}$-algebra. Write $\zeta_A(t)$ and $\zeta^{\lhd}_A(t)$ for its subalgebra and ideal zeta…

Let $h$ be a test function with Fourier transform $\hat h$, and let $\rho$ run over the zeros of the Riemann zeta function $\zeta(s)$. Resolvent trace identity. The global resolven…

Leading-term conjecture. The Taylor expansion of $Y_M(s)$ at $s=0$ satisfies

Truncation invariance conjecture. For every integer $k$ with $0\leqslant k<\operatorname{rk}(M)$,

Let $F:(\mathbb{C}^{n+1},\mathbf{0})\to(\mathbb{C},\mathbf{0})$ define a germ of an isolated hypersurface singularity. For $x\in F^{-1}(0)$, let $\mathcal{F}_{F,x}$ be the Milnor f…

Let $\mathcal{L}$ be a Calabi–Yau operator arising as a Picard–Fuchs operator for a Calabi–Yau motive $X_\varphi$. Let $p\geq5$, and let $\varphi$ be such that…