26 problems

Let $\mathcal{M}_g$ be the moduli space of smooth projective curves of genus $g$ over $\mathbb{C}$, let $\mathcal{A}_g$ be the moduli space of $g$-dimensional principally polarized…

Let $(\mathbb{V},Q)$ be a variation of polarized integral Hodge structures of even weight $2k$ on a smooth quasi-projective variety $S$. Let $\mathcal{A}_{0}(1,q)$ denote the degre…

Let $g$ be an integer with $g\geq 8$. Let ${\mathcal H}_g$ be the hyperelliptic locus in ${\mathcal M}_g$, let $j_h:{\mathcal H}_g\to {\mathcal A}_g$ be the restriction of the Tore…

Let $f:X_0\to\mathscr{S}_{\mathbf{I},F_{\infty}}^{\operatorname{ord}}$ be a locally closed immersion, so that $X$ is a subvariety of…

Let $(X,x)$, $X_0$, $f$, and $\T_{f,x}$ be as in the source, with $f:X_0\to X$ and with $\T_{f,x}$ the associated Tate-linear formal object. Let…

Zilber--Pink conjecture. If $V$ is an irreducible Hodge generic subvariety of $S$, then the intersection of $V$ with the special subvarieties of $S$ having codimension greater than…

Let $f:\mathcal{X}\to S$ be the family of abelian varieties considered in the paper, let $Z\subset S$ be an $\ell$-Galois special subvariety, let $x\in Z(\overline{\mathbb{Q}})$, a…

Let $S$ be the base of the family considered in the paper. An $\ell$-Galois special subvariety is a closed irreducible complex subvariety maximal among those having a fixed generic…

Let $S$ be the base of the family considered in the paper. A special subvariety is a closed irreducible complex subvariety maximal among those having a fixed generic Mumford–Tate g…

Let $S$ be defined over $\bar{\mathbb{Q}}$ and let $(\mathbb{V}^{\sigma})_{\sigma}$ be a $\bar{\mathbb{Q}}$-absolute variation on $S$. A closed irreducible subvariety is…

Let $(\mathbb{V}^{\sigma})_{\sigma}$ be an absolute variation of Hodge structure on a smooth irreducible quasi-projective complex algebraic variety $S$. For a closed irreducible su…

Let $S$ be a Shimura variety and let $V$ be a subvariety of $S$. For an irreducible subvariety $W$ of $S$, let $\langle W\rangle$ be the smallest special subvariety of $S$ containi…

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$. A special subvariety for $(S,{\mathbb V})$ is positi…

Let $S_\Gamma$ be the variety under consideration, and let bi-arithmetic points and bi-arithmetic subvarieties be defined as in the paper. The bi-arithmetic-point characterization…

Let $(G,X,\Gamma)$ be a real Shimura datum and let $W\subset S_\Gamma$ be an algebraic subvariety. The $\mathbb{Z}$-special-point characterization conjecture. The subvariety $W$ co…

Assume the rank-one setting, and let $\mathbb{Z}$-special points and $\mathbb{Z}$-special subvarieties of $S_\Gamma$ be defined as in the paper. Rank-one $\mathbb{Z}$-special-point…

Let $S_\Gamma$ be the variety under consideration, and let a $\Gamma$-special point and a $\Gamma$-special subvariety be defined as in the paper. The $\Gamma$-André–Oort conjecture…

Let $S_\Gamma$ be the smooth quasi-projective variety supporting the given variation of Hodge structure. For an irreducible subvariety $W\subset S_\Gamma$, Hodge codimension and Ho…

Zilber–Pink conjecture, finite-containment formulation. There is a finite collection $\Sigma$ of proper special subvarieties of $\mathfrak{S}$ such that every atypical subvariety…

Daw–Ren conjecture. There exists a finite set $\Omega:=\Omega(A)$ of semisimple $\mathbb Q$-subgroups of $\mathbf G$ such that

Let $S$ be an irreducible smooth quasi-projective variety endowed with a variation of mixed Hodge structures $\mathbb{V} \longrightarrow S$. Denote by…

Conjecture AB.

Conjecture B. The set of special points in $S$ is Zariski dense in $S$.

Let $X$ be an irreducible Mumford–Tate variety associated to a group $G$, and suppose that $X$ contains infinitely many special curves. For finitely many compactified special curve…

A mixed Shimura variety is an algebraic variety arising from a mixed Shimura datum; its special points and special subvarieties are the distinguished points and subvarieties define…