55 problems

Let $M$ be a $K$-motive, let $\Delta_K(M)$ be its fundamental line, and let $\vartheta_\infty:K_{\mathbb R}\otimes_K\Delta_K(M)\cong\mathbf{1}_{K_{\mathbb R}}$ be the period-regula…

For weight $n$ and depth $k$, let $P(n,k)$ denote the number of irreducible multiple Clausen values such that a rational basis at that weight and depth is formed from a minimum num…

Cube finite-place mean conjecture. The limit

Lang–Rohrlich conjecture. The value $\Gamma(z)$ is algebraic if and only if

Let $\zeta(n)$ denote the Riemann zeta value at the integer $n$, and let $\overline{\zeta}_m$ denote the symmetrized Mordell–Tornheim zeta value of order $m$. Rational-span conject…

Let $\overline{\zeta}_n$ denote the symmetrized Mordell–Tornheim zeta value of order $n$, and let $n\geq 1$. A value is a rational multiplicity of $\pi^n$ if it has the form…

Let $K={\mathbb F}_q(\theta)$, let $F$ be a finite extension of $K$, let ${\cal O}_F$ be an integral closure of $A$ in $F$, and let $F_{\infty}=F\otimes_K K_{\infty}$. Let…

Mazur–Tate leading-term conjecture. One has

Mazur–Tate conjecture. The following claims are valid:

Let $M$ be a motive over $K$, let $F=\mathcal{F}_\lambda$, and let $T$ be a stable $\mathcal{O}_F$-lattice in its realization. Assume the local hypotheses and the motivic cohomolog…

Let $M$ be a critical motive over $K$ with coefficient field $\mathcal{F}$, let $\Omega_{\gamma,\delta}$ be its period, and assume the height pairing $h_M$ is available. If $x$ and…

Let $M$ be a critical motive over $K$ with coefficient field $\mathcal{F}$. Let $\gamma$ and $\delta$ be $\mathcal{F}$-bases of ${\det}_\mathcal{F}(H_B(M)^+)$ and…

Let $y=(\kappa,\lambda)$ be an arithmetic, not necessarily crystalline, specialization, and let $?=\textup{\bf f},{\rm ad}$. The quantities…

Special-value conjecture.

Polynomial difference-equation conjecture. For each nonnegative integer $u$, the first equation has, up to multiplication by a nonzero constant, a unique polynomial solution $P$, a…

Let $F$ be totally real, let $H/F$ be a finite abelian CM extension with Galois group $G$, and let $S=R\cup\{\mathfrak{p}\}$, where $\mathfrak{p}$ splits completely in $H/F$. Let…

Let $p$ be a prime, let $k>1$ be an integer, and let the Hurwitz zeta function be … for $0<x\le1$. Milnor's conjecture. The real numbers … are linearly independent over…

Chowla–Chowla conjecture. $L(2,f)\ne0$, except when

Let $\Delta(\mathcal{M})$ be the fundamental line, let $\theta_\infty:\Delta(\mathcal{M})_\infty\overset\simeq\longrightarrow F_\infty$ be the comparison isomorphism, and let…

Let $\mathcal{M}$ be the modular motive, let $\Delta(\mathcal{M})$ be its one-dimensional fundamental line over $F$, and assume the non-degeneracy conjecture for the Gillet–Soulé h…

Equivariant Beilinson conjecture. The equality

Let $t^{\sfrac{1\!}{2}}(2,\{1\}^{2n+1},2)$ denote the multiple $t$-value with the indicated indices, and let $t(k)$ denote the ordinary $t$-value of weight $k$. Odd-weight evaluati…

Let … be Catalan's constant, and let $\mu(\beta(2))$ denote its irrationality measure. Irrationality-measure conjecture. The irrationality measure of Catalan's constant is … This i…

Let $\Pi$ be a RACAR of $\mathrm{GL}_3$, let $E$ be its coefficient field, and let $\eta$ be a Dirichlet character with Gauss sum $G(\eta)$. For $j\in\mathbb{Z}_{\geq 0}$, set … As…

Let $F$ be the totally real field with set of archimedean places $\Sigma_{\infty}$, and let $\pi$ and $\tau$ be the automorphic representations considered in the paper, with balance…