26 problems

Torsion Anomalous Conjecture. An irreducible subvariety $V$ of a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties.

Kahn's conjecture. If $r=s+1$, then the Somekawa K-group $K(k;G_1,\ldots,G_r)$ is torsion. If $r>s+1$ and $k$ does not admit an ordered field structure, then

Let $k$ be a field and let $G_1,\ldots,G_r$ be semi-abelian varieties over $k$. Let $K(k,G_1,\ldots,G_r)$ denote the Somekawa $K$-group, let $\mathcal{M}_k$ be a certain category o…

Let $B$ be a semi-Abelian variety over $\mathbb Q$, and let $W\subseteq B$ be an algebraic subvariety over $\mathbb Q$. Intersection conjecture for semi-Abelian varieties. There is…

Effective semi-abelian characterization conjecture. There exists an integer $k = k(d) \geq 1$ such that, whenever

Revised relative Manin–Mumford conjecture. If $Z(\mathbb{C})\cap\mathcal{G}_{\mathrm{tor}}$ is Zariski dense in $Z$, then either $Z$ is contained in a special subvariety of…

Revised relative unlikely-intersection conjecture. The intersection $Z\cap\mathcal{G}^{[>d]}$ is not Zariski dense in $Z$.

Pink's original relative Manin–Mumford conjecture. The intersection $Z\cap\mathcal{G}^{[>d]}$ is not Zariski dense in $Z$.

Pink–Mordell–Lang conjecture. If $Z$ is not contained in any translation of a proper algebraic subgroup of $G$, then $Z\cap(\Gamma+G^{[>d]})$ is not Zariski dense in $Z$.

Relative Manin-Mumford conjecture. The stated alternative holds.

Let $\mathcal{F}$ be an infinite set of $p$-tuples $f=(f_1,\ldots,f_p)$ of nowhere vanishing holomorphic functions on the disc $\mathbb D$ satisfying the identity referenced in the…

Let $E$ be an elliptic curve over $\overline{\mathbb{Q}}$, and let $G$ be an extension of $E$ by $\mathbb{G}_m$ parametrized by $Q\in E^*(\overline{\mathbb{Q}})$. Let $\Omega_G$ be…

Let $G$ be as in the relative semi-abelian conjecture, with notation $c_i$ and $t$ defined there. Explicit relative semi-abelian conjecture. … This is stated as a corollary of the…

Let $A$ be an abelian variety over $K$ of dimension $g$, and let $G$ be an extension of $A$ by $\mathbb{G}_m^s$ parametrized by $Q_1,\dots,Q_s\in A^*(K)$. Let $\Omega_G$ be its per…

Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$, and let $G$ be an extension of $A$ by the torus $\mathbb{G}_m^s$ parametrized by…

Let $G$ be a semi-abelian variety. Define a special subvariety to be a translate of a semi-abelian subvariety of $G$ by a torsion point, and call the zero-dimensional special subva…

Finite-rank Lehmer conjecture for products. There exists $c_\Gamma>0$ such that

Bounded Height Conjecture. For an irreducible variety $V$ in a semi-abelian variety, the set of maximal $V$-torsion anomalous points has bounded height.

Let $G$ be a semi-abelian variety and let $V$ be a weak-transverse variety in $G$. Let $V^{ta}$ denote the complement in $V$ of the union of all $V$-torsion anomalous varieties. Bo…

Let $G$ be a semi-abelian variety and let $V$ be a subvariety of $G$. A maximal $V$-torsion anomalous variety is a $V$-torsion anomalous variety not contained in any $V$-torsion an…

Let $G$ be a semi-abelian variety and let $V$ be a subvariety of $G$. A subvariety $Y$ is $V$-torsion anomalous if it is an irreducible component of $V\cap(B+\zeta)$, where…

Relative Lehmer conjecture. There is a real number $c>0$ such that every point $x\in G(\overline{\mathbf Q})$ not contained in any proper algebraic subgroup of $G$ satisfies

Zilber's conjecture. There is a finite family $\Phi$ of proper algebraic subgroups of $G$ such that, for every algebraic subgroup $H$ of $G$, every atypical component of $X\cap H$…

Pink's finite-rank conjecture. If $X$ is not contained in a translate of a proper algebraic subgroup of $G$, then

Pink's conjecture. If $X$ is not contained in a proper algebraic subgroup of $G$, then