10 problems

Colmez's conjecture. The identity

Let $A$ be a CM abelian variety of dimension $d$, with CM field $E$ Galois over ${\bf Q}$ and $G={\rm Gal}(E|{\bf Q})$. Let $\Phi$ be the characteristic function of the CM type and…

Refined Colmez conjecture. There exists an integer $m\geq1$ and an element $\beta=\beta(A,m,\phi)\in\omega_{\cal A}^{\otimes m}$ such that (a) $\beta$ generates…

Let ${\mathcal{CM}}^0$ be the space of class functions associated with CM-types, and let $ht$ and $Z(\mathord\cdot,s)$ be the linear height and logarithmic-derivative functionals d…

Let $A$ be a CM abelian variety of dimension $g\geq 1$, let $[o]\in\mathbb{A}_g(\overline{\mathbb{Q}})$ parametrize $A$, and set $[\mathbf{o}]:=([o],\ldots,[o])$. An elementary qua…

Let $A$ be a square-free abelian variety of dimension $g$ with complex multiplication by the CM algebra $E={\rm End}(A)\otimes {\mathbb Q}$, and let $\theta_1,\ldots,\theta_g$ be i…

Let $X$ be a CM abelian variety over a number field $K$, let $E$ be its CM algebra with CM-type $\Phi$, let $H_E=\operatorname{Hom}_Q(E,Q^{\rm alg})$, and assume the notation…

Let $A$ be the CM abelian variety in the source, let $g=\dim A$, let $\mathbb{T}_{\mathfrak{P}}(A)$ be its $p$-adic representation, and let…

Let $K$ be a number field and let $A$ be an abelian variety with complex multiplication by $K$. Periods of $A$ are the complex numbers obtained by comparing suitable algebraic de R…

Let $A$ be a CM abelian variety over $\mathbb{Q}^{\mathrm{al}}$, with good reduction $A_0$ over $\mathbb{F}$. A Hodge class on $A$ is an element of $H_{\mathbb{A}}^{2*}(A)(*)$, and…