33 problems

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…

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

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…

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…

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 $X$ 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…

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…

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 $k$ 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…

Let $\alpha\in{\mathbb R}\setminus\{0,1\}$, let $X_\alpha$ be the associated smooth compactification, and let $\Delta_\alpha^+=\frac12(F_\infty+1)\Delta_\alpha$ be the real-Frobeni…

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…

Let $L\subset\mathbb{C}$ be a number field with ring of integers $\mathcal{O}_L$, let $X/L$ be a smooth projective variety equipped with a toroidal embedding $Z\subset X$, and let…

Let $M(n,1)$ denote the maximum value of the regulator-related function for $r_2=1$, and let $C(1)$ be the corresponding threshold. Maximizer conjecture for $r_2=1$. One has…

For each degree $n$ and signature parameter $r_2$, let $M(n,r_2)$ denote the maximum value of the corresponding function $Q(n,r_2,\{j_1,\ldots,j_{r_2}\},\cdot)$, with the conjectur…

Let $Q(5,1,\{4\},(x,y,1,g))$ be the function obtained after reducing to $z=1$ and $g\neq\pm1$, and let $(x,y,z,g)$ denote its parameters. Maximum-value conjecture. The maximum of…

Toric regulator graph construction conjecture. The projected toric regulator equals the graph-theoretic regulator:

Sreekantan regulator factorization conjecture. This composition factors via the Sreekantan regulator.

Let $L/K$ be a finite Galois extension of number fields with group $G$, let $S$ contain all archimedean places, and let $r>1$ be an integer. For a complex character $\chi$ of $G$,…

Let $f$ be a normalized newform, $F$ the number field in the paper, and $f_F$ its base change. Let $\widetilde{u}_2(f_F)$ be the normalized degree-two period quantity, $\Omega_f^+$…

Let $d$ be the degree parameter of the family considered in the source, let $X_t$ be the corresponding fiber, and let $\mathscr{I}_0$, $\mathscr{I}_{\infty}$, and $\mathscr{D}$ be…

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…

Let $\psi\in\operatorname{Irr}_{\mathbb C_p}(\mathcal G)$, choose $\iota:\mathbb C\simeq\mathbb C_p$, and let $r_S(\psi)$ be the corresponding complex order of vanishing. Let…

Gauss regulator equality conjecture. The first equality in Theorem is valid even when $a=b$.