331 problems

Let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction $A_0$ over $\mathbb{F}^{\mathrm{al}}$, let $d=\dim(A)$, and let $\gamma$ be an absolute Hodge clas…

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

Let $(A,\Theta)$ be a principally polarized abelian variety of dimension $g$, and let $X\subset A$ be a subvariety of dimension $d$ with $1\le d\le g-2$. The conjecture asserts tha…

Let $\mathcal{S}$ be a split unitary Shimura variety of signature $(g,g)$ at a split prime $p$, and let $\mathcal{S}_{\mathrm{basic}}$ denote its basic locus. Outside the degenerat…

Let $\pi:\mathcal{A}\to S$ be an abelian scheme over an algebraically closed field of characteristic $0$, and let $X\subseteq\mathcal{A}$ be an irreducible closed subvariety. If th…

Let $\mathcal{A}_g$ be the moduli space of principally polarized complex abelian varieties of dimension $g$, let $T_g$ be the Torelli locus, and let $T_g^o$ be its open Torelli loc…

Rasmussen–Tamagawa conjecture. The set $\mathcal{H}(K,g)$ is finite for any choice of $K$ and $g$. This strengthens the known finiteness of $\mathcal{H}(K,g,\ell)$ when the prime…

Let $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$. For each prime $\ell$, let $G_{A,\ell}$ be the Zariski closure of the image of the $\ell$-adic represe…

Let $(A,\theta)$ be a polarized abelian variety, and let $C\subset A$ be a smooth curve of genus $h$. Write $c$ for the gonality of $C$ and … for its $\theta$-degree. Gonality–genu…

Geometric Bogomolov conjecture. If $X$ has dense small points, then $X$ should be special.

Let $k\in\mathbb{Z}_{>0}$, let $A$ be a very general abelian variety, and let $\mathscr{G}(k)$ be the least dimension such that a very general abelian variety of that dimension has…

Let $A$ be an abelian variety of dimension $g$ defined over $\mathbb{Q}$. For each prime $p$ of good reduction, write the normalized Euler factor as … Let…

David-Hindry's conjecture. For every integer $n\in\mathbb{N}$, there is a constant $c(A/K,\mathcal{L},n)>0$ such that, for every $n$-tuple $(P_1,\ldots,P_n)$ of points of infinite…

Let $K$ be a totally real field and $L$ a number field. Let $A/L$ be an abelian variety of $\mathrm{GL}_2$-type such that … For a prime $\mathfrak{p}$ of $K$ above a rational prime…

Let $A$ be an abelian variety over $\mathbb C$ of dimension $g$. Its Beauville decomposition is … where … Here $n^*$ is induced by multiplication by $n$ on $A$. Beauville's conject…

Pareschi–Popa conjecture. If $\Theta=X+Y$ decomposes nontrivially as a sum, then either $(A,\Theta)$ is the Jacobian of a smooth projective curve, or $(A,\Theta)$ is the intermedia…

Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. Write $\langle\mathrm{H}^1_{\mathrm{dRB}}(A,\mathbb{Q})\rangle^{\otimes}$ for the tensor category generated by its first…

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 $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction at $w$, and let $\gamma$ be a Hodge class on $A$. Call $\gamma$ $w$-Lefschetz if its specializatio…

Let $p>0$, let $g$ be a positive integer, and let $x=\left[\left(A_x,\lambda_x\right)\right]$ be a $\overline{\mathbb{F}}_p$-point of the moduli space $\mathcal{A}_g$ of $g$-dimens…

Let $A$ be an abelian variety defined over a number field $K$, and let $\mathcal{L}$ be a symmetric ample line bundle on $A$. For a point $P\in A(\overline{K})$, let $D=[K(P):K]$ a…

Ueno's Conjecture K strengthening. There exists an isogeny $A' \to A$ such that

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 $K$ be a finitely generated field over its prime field, let $e$ be a positive integer, and let $\sigma$ range over $\mathrm{Gal}(K^\mathrm{sep}/K)^e$. For almost all such…

Ahmadi–Shparlinski's conjecture. Every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank.