9 problems

The Tate conjecture. There are finitely many isogeny classes of abelian schemes over $C$.

Let $A\in \mathcal{A}_{g,1}({\mathbb F}_{q})$ be a geometrically simple principally polarised abelian variety with Weil $q$-number $\alpha_A$, and suppose that its $p$-divisible gr…

Conjecture on isogeny class sizes. We have

Let $K$ be a number field, $S$ a set of finite places of $K$, and $g\geq 2$ an integer. Let $\mathcal{F}$ be a one-parameter family of complex abelian varieties $A$ of dimension…

Asymptotic simplicity conjecture. The proportion of isogeny classes that are simple should converge to one:

Principally polarized isogeny-class size conjecture. For a positive proportion of integers $n$, their number is

Let $R_g^{(p)}$ denote the set of Picard numbers of $g$-dimensional abelian varieties in characteristic $p>0$. Let $g \geq 1$ and let $\rho \in R_g^{(p)}$. Countable-isogeny-class…

Let $T=S\times A$ be the product of a Shimura variety $S$ and an abelian variety $A$, and let $X\varsubsetneq T$ be a weakly special generic closed irreducible subvariety. Fix a su…

In the setting of the preceding results, let $S$ be a Shimura variety, let $A$ be an abelian variety, let $\operatorname{CM}\subset S(\overline{\mathbb{Q}})$ denote the special poi…