15 problems

Let $M$ be a $3$-manifold, let $n$ be a positive integer, and let $\rho$ be a boundary-unipotent representation of $\pi_1(M)$ in $\mathrm{SL}(n,\mathbb C)$ or…

Let $F$ be a subfield of $\mathbb{C}$ with embedding $\sigma:F\hookrightarrow\mathbb{C}$. Let $\mathcal B(F)$ and $\mathcal B(\mathbb{C})$ denote the corresponding Bloch groups, an…

Let $E$ be a cubic curve in $P^2$. For a point $p\in P^2$ and three lines $l_1,l_2,l_3$ through $p$, let $A_i:=l_i\cap E$ and define … Let $\tilde R^*_3(E)\subset I^4_E$ be the sub…

Let $F$ be an infinite field. Let $B_3(F)$ be Goncharov's Bloch group defined using the stated relations, and let $\mathcal{L}^{\mathrm{f}}_3(F)$ be the weight-three component of t…

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $\mathcal{G}_k$ be the quotient of $\operatorname{gr}_k^\mathcal{D}\mathcal{L}_{2k}(F)$ by the subs…

Let $F$ be the field under consideration, let $k$ be a positive integer, let $x_2,\ldots,x_k\in F^\times$, and let $w_0,\ldots,w_4\in\mathbb{P}^1(F)$. For each $i$, omit $w_i$ from…

Let $\mathcal{M}$ and $\mathcal{N}$ be hyperbolic $3$-manifolds. Write $\operatorname{vol}_{\mathbb{C}}$ for complex volume and $\beta$ for the Bloch invariant. Ramakrishnan's conj…

Modified Bloch group conjecture. The homomorphism $\psi_{F,\mathrm{tf}}$ is an isomorphism. Extensive computations for imaginary quadratic fields motivate the claim that the modifi…

Let $F$ be a field and let $R_2(F)$ be the weight-two relation subgroup in the modified definition using $F\setminus\{0,1\}$. An inverted five term relation is a relation obtained…

Let $F$ be a field and let $R_2(F)$ be the subgroup of the relevant free abelian group generated by functional relations in weight two. For $x,y\in P^1_F$, write $FT[x,y]$ for the…

Strong reciprocity conjecture. The residue map $res_{|C|}:\Gamma(k(C),3)\to\Gamma(k,2)$ is homotopic to zero. More precisely, there exists a map $h$ fitting into a commutative diag…

Let $F \subset \mathbb{C}$ be a number field not contained in $\mathbb{R}$, and let $\mathcal{M}$ be the set of manifolds with invariant trace fields contained in $F$, … Let…

Let $F\subset\mathbb C$ be a concrete number field which is not contained in $\mathbb R$. Let $\mathcal B(F)$ be the Bloch group of $F$, and call a hyperbolic $3$-manifold admissib…

Let $A=(a_{ij})$ be a positive definite symmetric $r\times r$ matrix with rational entries. Consider the system … For a solution $\mathbf{x}=(x_1,\ldots,x_r)$ in a number field $F$…

Let $A$ be a positive definite symmetric $r\times r$ matrix with rational coefficients. Let $(Q_1,\dots,Q_r)$ range over the solutions in number fields of the algebraic system … ma…