127 problems

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…

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

Manin–Mumford conjecture. The subvariety $X$ is a translate of an abelian subvariety.

Pink's conjecture. The intersection

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…

Let $f,g$ be polynomials of the same degree $d\geq 2$, and let $\operatorname{Prep}(f)$ denote the set of preperiodic points of $f$. DeMarco–Krieger–Ye's conjecture. There should e…

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…

André–Pink conjecture. If $V$ has Zariski dense intersection with $O$, then $V$ is totally geodesic.

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…

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

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…

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…

Density conjecture for the typical Hodge locus. If $\operatorname{HL}(S, {\mathbb V}^{\otimes})_{\operatorname{typ}}$ is not empty, then it is dense in $S$ for the analytic topolog…

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…

Large Galois orbits conjecture. There exist positive constants $C_1$ and $C_2$ such that, for every $s\in C\cap\Sigma$,

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…

Defect-condition conjecture. The defect condition holds in all mixed Shimura varieties. The paper notes that this condition is known for products of a pure Shimura variety and an a…

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 $\mathbb{H}$ be the complex upper half-plane and let $J=(j,j',j”):\mathbb{H}\to\mathbb{C}^3$, extended coordinatewise to $\mathbb{H}^n$. A subset $U\subseteq\mathbb{H}^n$ is…

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 $A$ 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…

Subquadratic growth conjecture. The quantity $\log|A_N|$ satisfies $\log|A_N|=o(N^2)$; more strongly, $\log|A_N|=\mathcal{O}(N\log N)$.

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…