31 problems

André-Oort conjecture. If $V$ contains a Zariski dense set of special points, then $V$ is a Shimura variety.

For $x otinboxminus$ a special point of the moduli space $mathcal{A}g$ of principally polarized abelian varieties, let $Ax$ be the corresponding $g$-dimensional principally polariz…

Let ${\mathcal X}_{\mathbb C}$ be the complex universal abelian variety considered in the paper, let $Y\subseteq {\mathcal X}_{\mathbb C}$ be an irreducible subvariety, and let…

Let $\mathbf{I}$ be a finite index set, let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ separate factors over $\mathscr{A}_{g_{\mathbf I},\mathbb{F}}$, and for each $i\in\mathbf I$…

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ be a locally closed immersion with image in the ordinary locus. A subvariety is quasi-weakly special when its ordinary locus is conta…

Dynamical André-Oort conjecture. The following are equivalent:

Let $(M,X_M)$ and $(G,X)$ be Shimura data, let $x\bin X_M$ be Hodge generic, and let $\bSigma$ be the image in $Sh_K(G,X)$ of the hybrid orbit of $(M,X_M,x)$. Let…

Let $\Gamma\backslash X^+$ be an arithmetic variety, let $V$ be an irreducible algebraic subvariety, let $(M,X_M)$ be a Shimura datum, and let $x\in X_M$ be Hodge generic. Write…

Let $\Gamma\backslash X^+$ be an arithmetic variety, let $V$ be an algebraic subvariety, and let $\{Z_1;\ldots\}$ be the special subvarieties $Z_i\subseteq V$. Density formulation…

Let $\Gamma\backslash X^+$ be a Shimura variety, let $V$ be an algebraic subvariety, and let $\{Z_1;\ldots\}$ be the set of all special subvarieties…

Let $f:X_0\to\mathscr{S}_{\mathbf{I},F_{\infty}}^{\operatorname{ord}}$ be a locally closed immersion, and let $X$ be its image. Suppose that $X$ possesses a Zariski-dense collectio…

Let $F_{\infty}$ be an algebraic closure of F_{\infty}_p, and let $X\subseteq\mathcal{A}_{g,F_{\infty}}$ be a generically ordinary subvariety containing a Zariski-dense collectio…

Let $\Gamma\subset G(\mathbb Q)$ be an arithmetic subgroup, let $S=\Gamma\backslash\Omega$, and let $V\subset\mathbb C^{N}\times S$ be an irreducible algebraic variety. Let…

Let $k$ be a number field, let $S$ be a finite set of places of $k$ including all the archimedean places, and let $X$ be a special subvariety of a Shimura variety, defined over $k$…

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

Let ${\mathbb V}$ be a ${\mathbb Z}$-variation of Hodge structure on a smooth irreducible complex quasi-projective variety $S$, with generic Hodge datum $({\pmb G},{\mathcal D})$.…

Let $(G,X,\Gamma)$ be a real Shimura datum and let $W\subset S_\Gamma$ be an irreducible algebraic subvariety. A CM-point is a point of $S_\Gamma$ at which the Mumford–Tate group o…

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…

Effective André–Oort-type conjecture. There are effectively computable constants $c_i=c_i(V)$ such that whenever

Collinear triples conjecture. There are only finitely many triples with the $P_i$ pairwise distinct and lying on a straight line which is neither horizontal, vertical nor the diago…

Let $V\subsetneq\mathbb{C}^{3n}$ be an algebraic variety. A $J$-special subvariety is an irreducible component of the Zariski closure of the image under $J$ of an $\mathbb{H}$-spec…

Let $T=S\times A$ be the product of a Shimura variety $S$ and an abelian variety $A$. A subvariety of $T$ is weakly special generic if it is not contained in any smaller weakly spe…