92 problems

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…

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 $Y(1)$ 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…

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…

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…

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

Let $n$ 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 $f_t(x)$ be a non-isotrivial algebraic family of rational functions of degree greater than $1$ over $\mathbb C$, parametrized by $t\in\mathbb A^1_{\mathbb C}$, and let…

Large Galois Orbits conjecture. There exist positive constants $C_{\mathrm{mult}}$ and $C_{\mathrm{exp}}$ such that, for all $s\in C\cap\Sigma$,

Let $S$ be a Shimura variety and let $V$ be an irreducible subvariety of $S$. An optimal point is an optimal subvariety of $V$ of dimension zero, and write…

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 $S$ be a Shimura variety and let $V$ be an irreducible subvariety of $S$. An irreducible subvariety $W$ of $V$ is called optimal if, for every irreducible subvariety $Y$ of $V$…

Let $S$ be a Shimura variety and let $V$ be an irreducible subvariety of $S$. An irreducible subvariety $W$ of $V$ is called atypical if it is an irreducible component of $V\cap Z$…

Pink's conjecture. The intersection

Let $X \stackrel{\iota}{\hookrightarrow} {\mathcal S}$ be a closed irreducible algebraic subvariety of a mixed Shimura variety ${\mathcal S}$. Let…

Let $\mathcal{Y}\to\mathbb{P}^1$ be a Lefschetz pencil of degree-$d$ surfaces in $\mathbb{P}^3$. Picard-number finiteness conjecture. If $d\geq 5$, the Picard number of…

Real multiplication finiteness conjecture. Fix $g\geq 7$. Each $E_{g,K}$ is contained in a finite union of Hilbert modular varieties. Moreover, $E_{g,K}$ is empty for all but finit…

Let $\mathcal{A}\rightarrow S$ be the abelian scheme in the paper, and let $\mathcal{X}\subseteq\mathcal{A}$ be a subvariety. For $t\in\mathbb{N}$, let $\mathcal{A}_{(\leq t)}$ be…