92 problems

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…

Zilber's atypical-intersection conjecture. The subvariety $V$ contains at most finitely many maximal atypical subvarieties.

Let $\mathcal{Y}$ be an irreducible subvariety of ${\bf G}_m^n$ defined over ${\bf C}$. For algebraic subgroups $H\subset {\bf G}_m^n$ satisfying … consider the union of the inters…

Let $Y(1)$ denote the modular curve, and let $S\subset Y(1)^n$ be an irreducible curve that is not contained in a proper special subvariety of $Y(1)^n$. For a special subvariety…

Pink's conjecture. The intersection

Let $n$ be a positive integer and let $V\subseteq\mathbb{A}_{\mathbb{C}}^{n}$ be an algebraic variety. For a subvariety $X\subseteq V$, define its defect by … The subvariety $X$ is…

Let $E_1$ and $E_2$ be elliptic curves defined over $\mathbb{C}$, given by Weierstrass equations … where $f_1$ and $f_2$ do not have the same roots. Write…

Let $X$ be a variety and let $\Phi:X\to X$ be a dominant map. A point of $X$ is preperiodic for $\Phi$ if its forward orbit under $\Phi$ is finite, and a subvariety $Y$ is preperio…

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

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}^{\mathrm{ord}}$ be a map from a smooth connected variety, let $x\in X(\mathbb{F})$, and let…

Let $S$ be a Shimura variety and let $V$ be an irreducible subvariety of $S$. André–Pink–Zannier conjecture. Suppose that the intersection of $V$ with a single Hecke orbit in $S$ i…

Let $\Gamma\backslash X^+$ be a Shimura variety, let $V$ be an algebraic subvariety of it, and let $x\in X^+$. Write ${\mathcal{H}}_{\Gamma\backslash X^+}(x)$ for the generalised H…

Let $\pi: \mathfrak{A}_{g,l} \to A_{g,l}$ be the universal family of complex principally polarized abelian varieties of dimension $g$ with symplectic level $l$-structure, and let…

Let $M$ be a mixed Shimura variety, let $Z\subset M$ be a subvariety, and let $\delta(A)=\dim\langle A\rangle-\dim A$ denote the defect of a subvariety $A$, where…

Modified André–Pink–Zannier conjecture. If $\mathcal{A}_{\Gamma}\cap\mathcal{V}$ is Zariski dense in $\mathcal{V}$, then one of the following holds:

Let $V\subseteq (\mathbb{C}^{\times})^n$ be an algebraic variety. Atypical subvarieties are the components of intersections with algebraic subtori having larger-than-expected dimen…

Let $G$ be a group variety and let $V\subset G$ be irreducible. A maximal $V$-torsion anomalous point is a point that is a maximal $V$-torsion anomalous variety under the terminolo…

Let $A$ be an abelian variety of dimension $g$, let $V\subset A$ be an irreducible subvariety of dimension $d$, and let $\Gamma\subset A$ be a subgroup of finite rank. The variety…

Generalized relative Manin–Mumford conjecture. Under these hypotheses, $Y$ is contained in a proper torsion coset.

Let $S$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$, let $\Phi:S\times\mathbb{P}^N\to S\times\mathbb{P}^N$ be an algebraic family of endomorphisms of degree…

Unlikely intersection bound. For every $\varepsilon>0$, there exists $c=c(n,\mathcal{V},\varepsilon,K)\in\mathbb{R}$ such that, for every flat subgroup scheme…

Mixed hybrid conjecture. (1) The subvariety $S\subseteq {\mathbb A}_\Gamma$ is weakly special. (2) If, for some $s\in S$, one has $V\cap (A_s)_{\operatorname{tors}}\neq\varnothing$…

Let $S$ be a smooth irreducible quasi-projective variety over ${\mathbb C}$, let $K$ be a positive integer, and let … be a family of endomorphisms with $\Phi(s,p)=(s,F_s(p))$, wher…

Let $G/\mathbb{Q}_p$ be a reductive group, let $[\mu]$ be a minuscule conjugacy class of cocharacters of $G_{\overline{\mathbb{Q}}_p}$, and let $b\in G(\breve{\mathbb{Q}}_p)$ repre…

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…