104 problems

Let $u$ be a positive integer congruent to $4$ modulo $8$, such that $\operatorname{ord}_p(u(u^2-1))$ is zero or odd for every odd prime $p$. Assume conditions (1), (3), and (4) of…

Mazur–Tate conjecture. The following claims are valid:

For positive integers $w\geq 2$ and $d\geq 1$, let $N_{w,d}$ be the number of primitive multiple Clausen values of weight $w$ and depth $d$. Broadhurst's dimension conjecture. The…

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

Special-value conjecture.

Let $X_p$ be the $p$-adic symmetric space attached to an orthogonal group, and let $J_{\mathscr{D}}$ be a rigid meromorphic cocycle that can be evaluated at a suitable special poin…

Let $V\cong\mathcal K^{2d}$ be an even-dimensional $p$-adic Galois representation of $G_{\mathbb Q}$ satisfying condition $\mathrm{(Geom)}$, with no Artin representation as a subqu…

Let $F$ be totally real, let $C_{\mathfrak f}$ be the narrow ray class group modulo $\mathfrak f$, and let $s_\iota$ denote complex conjugation classes at the real embeddings. Let…

Let $n$ be an integer with $n\geq 0$, and write $\zeta(\{a\}^n)$ for the multiple zeta value with $n$ successive arguments equal to $a$. A multiple zeta value is called self-dual w…

Let $K$ be a real quadratic field, let $\mathfrak{C}$ be a narrow ideal class of $K$, and define … where $\mathfrak{C}_1$ and $\mathfrak{C}_2$ are the ideal classes used in this co…

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…

Let $X$ be in the function-field setting, let $q$ be the cardinality of the constant field, and suppose the integral regulator maps have finite kernel and cokernel with orders…

Let $P(n)$ be the emptiness formation probability in the thermodynamic-limit antiferromagnetic XXX Heisenberg spin chain, defined by … where $P_j=S^z_j+\frac12$ projects the $j$-th…

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…

Let ${\cal C}(n):=\{\operatorname{mcl}(a_1,\dots,a_k):a_1+\dots+a_k=n\}$ be the set of multiple Clausen values of weight $n$. Let $I(n)$ be the size of a smallest rational basis fo…

Cube finite-place mean conjecture. The limit

Let $p$ be a prime, let $\Lambda$ be a commutative pro-$p$-finite ring, and let $M$ be a lisse $\Lambda$-sheaf on $\operatorname{Spec}(\mathbb{Z}[1/S])_{\mathrm{et}}$, where $S$ is…

Let $E_u$ be the elliptic curve under consideration, let $R_u$ be the regulator obtained from the integral motivic lattice, and for each finite prime $\ell$ let $T_\ell$ and…

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

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…