29 problems

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 an arithmetic variety, let $V$ be an algebraic subvariety, and let $\Sigma$ be the set of special points of $\Gamma\backslash X^+$ that belong to $V$.…

André–Oort conjecture in genus zero. The variety $S$ is $\overline{\mathbb Q}$-bi-algebraic. Furthermore, if not all residues are identically zero on $S$, then the fibers of the re…

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 $\Gamma\subset G(\mathbb Q)$ 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…

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…

Let $Y\subset {\rm Sh}_K^0({\mathbf P},X)$ be a closed irreducible algebraic subvariety of a connected mixed Shimura variety. Classical André–Oort conjecture. If $Y$ contains a Zar…

Equidistribution conjecture. Under the stated finiteness condition, the Galois-orbit measures $\mu_i$ converge weak-$*$ to the uniform probability measure on $X$.

Let $\mathcal{H}$ be the upper half-plane, and let $\tau_1,\dots,\tau_n\in\mathcal{H}$ be quadratic points lying in distinct $\operatorname{GL}_2^+(\mathbb{Q})$-orbits, with none l…