16 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…

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 $M$ 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…

Large Galois Orbits conjecture. There exist positive constants $C_{\mathrm{mult}}$ and $C_{\mathrm{exp}}$, depending only on $g$, $L$ and $C$, such that for every point…

Let $S$ be a parameter space mapping to $\Gamma \backslash D_{S}$, let $(\mathbf{M},D_{M})\subset(\mathbf{G}_{S},D_{S})$ be a Hodge subdatum, and let $(2)$ denote the possibility t…

Let $V'\subsetneq\Gamma\backslash X^+$ be an irreducible subvariety, and let $ZP_{V',W}$ denote the union of the atypical intersection components defined in the paper using…

Let $\textnormal{S}_{\alpha}$ be a stratum of abelian differentials up to scaling. An arithmetic point has a degree, and points of degree at least $3$ are those whose arithmetic de…

Rémond's conjecture. The following forms are expected to hold:

Strong Zilber–Pink equivalence. The following conditions are equivalent: (a) the atypical Hodge locus is a finite union of maximal atypical special subvarieties; (b) it is a strict…

Let $K$ be an algebraically closed field and let $\mathfrak{C}$ be a distinguished category over $K$. A distinguished variety is an object of $\mathfrak{C}$, and…

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…

Habegger–Pila optimality conjecture. The subvariety $V$ contains at most finitely many optimal subvarieties of defect at most $d$.

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

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

Let $M$ be a Shimura variety. For each integer $d$, let $M^{[d]}$ be the union of Shimura subvarieties of codimension at least $d$. Let $V\subsetneq M$ be a Hodge generic closed ir…

Let $E_\lambda$ be the Legendre family of elliptic curves, and let $\mathcal{C}$ be an irreducible curve in an $n$-fold fibered power of this family, defined over the algebraic num…