60 problems

Let $X$ be an equidimensional compact complex algebraic manifold. Write $K_n(X)_{coperatorname{\mathbb{Q}}}$ for rational $K$-theory, $CH^{p}_{s}(X,n)_{coperatorname{\mathbb{Q}}}$…

Bertrand–Rodriguez Villegas conjecture. There exist absolute constants $c_0>0$ and $c_1>1$ such that, for every number field $L$, every $j\in\mathbb{Z}_{>0}$, and every nonzero

Let $X_{{\mathbb Q}}$ be a smooth projective variety over ${{\mathbb Q}}$, and let $X_{{\mathbb R}}:=X_{{\mathbb Q}}\times_{{\mathbb Q}}{{\mathbb R}}$. Let…

Hodge-$\mathcal D$-Conjecture. The map $r_{\mathcal{D},\mathbb{R}}^{p,n}$ is surjective. This is a higher-dimensional and higher-cycle analogue of the Hodge conjecture, formulated…

Stark's conjecture. The quotient $R_{K_0}/R_K$ is the determinant of an $(mn-n)\times(mn-n)$ matrix. This matrix should be diagonalized into $m-1$ blocks, each an $n\times n$ matri…

Let $K^+/k$ be the totally real subextension under consideration, let $G^+=\operatorname{Gal}(K^+/k)$, let $S_0=S_{\rm ram}(K/k)$, and let $U_{S_0}(K^+)$ be the group of global…

Weight-four inverse map conjecture. There exists a map

Beilinson–Cheeger–Chern class conjecture. For all $p \geq 1$ and $n \in {\mathbb N} \cup\{\infty\}$,

Let $C$ be a smooth projective geometrically irreducible algebraic curve of genus $g$ over a number field $k$. Let $H^2_{\mathscr{M}}(C,\mathbb{Q}(2))_{\mathbb{Z}}$ be the integral…

Weak mixed linear independence conjecture. As ${\mathbb Q}$-subspaces of ${\mathbb R}$,

Canonical splitting conjecture. There is a canonical direct-sum decomposition

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 $\mathcal{B}_0$ be the modular curve containing a CM point $\mathfrak{b}_0$, let $\mathcal{C}$ be its quotient containing the image $\mathfrak{c}$ of $\mathfrak{b}_0$, and let…

Toric–Sreekantan regulator conjecture. For each prime $\ell$, the valuation of the toric regulator at $\ell$ is the Sreekantan regulator tensored with ${\mathbb Q}_\ell$. This conj…

Let $R(N)$ denote the regulator associated with the real quadratic field determined by $N$. Large-regulator conjecture. There exists an infinite set of values of $N$ such that … Th…

Let $\mathcal{C}^{\underline{a}}_{m,n}$ be the family of curves cut out by $F^{\underline{a}}_{m,n}$, and let $R_{\gamma_1}(\underline{a})$ be the regulator period asymptotic to…

Let $X$ be a smooth projective variety, and let $\underline{CH}^{k}(X,m)$ be the relevant higher Chow group, $\underline{r}_{k,m}$ its regulator, and $D(k,m)$ the indicated denomin…

Assume that $k$ is totally real, $K/k$ is a CM Galois extension with group $G$, $c\in G$ is complex conjugation, and $V$ is a totally odd representation of $G$ over $\mathbb C_p$.…

Let $K/k$ be a Galois extension with group $G$, let $X$ be the degree-zero subgroup of the free abelian group on the Archimedean places of $K$, and let $A^*$ be the unit group of…

Let $K/k$ be a Galois extension with group $G$, let $V$ be an irreducible complex representation of $G$, and let $X$ be the degree-zero subgroup of the free abelian group on the Ar…

Let $K$ range over real quadratic fields, with regulator $R_K$, discriminant $d_K$, and parameter $m_K$ as used in the source. Regulator-growth conjecture. There exists an infinite…

Iterative upper-bound conjecture. For the $r_2=1$ case,

Let $X$ be an arithmetic scheme such that $X_{\mathbb{C}}$ is smooth and quasi-projective, and let $n<0$. Let $G_{\mathbb{R}}=\operatorname{Gal}(\mathbb{C}/\mathbb{R})$, let…

Let $M_\pi$ be the adjoint motive described above, let $r_p$ be its $p$-adic regulator and $r_\infty$ its archimedean regulator, and let $K$ be the relevant coefficient field. Beil…

Let $k$ be a number field, let $S_{\infty}$ be the set of its archimedean places with $|S_{\infty}|\geq 2$, and let $\Lambda_{S_{\infty}}(k)$ be the logarithmic embedding, of rank…