Rotger: Shimura varieties

Let $\overline{k}$ be an algebraically closed field, $A$, $\mathcal{L}/\overline{k}$ a polarized abelian variety. We are interested in:

1. Field of moduli of $(A, \mathcal{L})$ and field of moduli of ${\mathrm{End}}(A)$
2. Fields of definition of $(A, \mathcal{L})$ and fields of definition of ${\mathrm{End}}(A)$.

The field of moduli of $(A, \mathcal{L})$ $\mathcal{K}_{A, \mathcal{L}}$ is the unique minimal field in $\overline{k}$ such that $(A, \mathcal{L})^\sigma \iso (A, \mathcal{L})$ for all $\sigma \in {\mathrm{Gal}}(\overline{k}/\mathcal{K}_{A, \mathcal{L}}$).

$$\xymatrix{&&\mathcal{K}_{\mathrm{End}(A)}\\ & \mathcal{K}_S \ar@{-}[ur]& \\ \mathcal{K}_{(A, \mathcal{L})}\ar@{-}[ur] &&}$$

$\mathbb{Z} \subset S \subset {\mathrm{End}}(A)$. The top field is the unique field fixed by all $\sigma$ such that there exists $A \iso A^\sigma$ making the following diagram commute:

$$\xymatrix{A \ar[r]\ar[d]_{\beta \in {\mathrm{End}}(A)} & A^\sigma \ar[d]^{\beta^\sigma}\\ A \ar [r] & A^\sigma }$$

Shimura: the generic odd dimensional polarized abelian variety admits a model over $\mathcal{K}_{A, \mathcal{L}}$. The generic even dimensional polarized abelian variety does not admit a model over $\mathcal{K}_{A, \mathcal{L}}$.

(Silverberg) Fix $\mathcal{K}_{A, \mathcal{L}}$. There is a (unique, Galois) minimal field of definition $\mathcal{K}_S/K_{(A, \mathcal{L})}$ of $S \subset {\mathrm{End}}(A)$.

(Silverberg) There is a $H_{d,r}$ such that $\vert{\mathrm{Gal}}(K_S/K_{A, \mathcal{L}})\vert \leq H_{d,r}$ for any abelian variety $A$ such that $\dim A = d$ and $S \subset {\mathrm{End}}(A)$ with $[S:\mathbb{Z}] = r$.

If $A$ is simple, ${\mathrm{End}}(A)$ is an order in either a totally real field, a division algebra over a CM-field, or a quaternion algebra.

We will focus on the latter case.

Forgetful maps between Shimura varieties and rational points

Let $F$ be a totally real number field, with $[F:\mathbb{Q}] = n$. Let $B$ be an indefinite quaternion algebra over $F$ (that is, $B \tensor_{\mathbb{Q}} \mathbb{R} \iso M_2(\mathbb{R})^n$). Let $\mathcal{O}_B$ be a maximal order in $B$. Let

$$\Gamma_B = \{ \gamma \in \mathcal{O}_B: N_{B/F}(\gamma) = 1\} \subset SL_2(\mathbb{R})^n.$$

Moduli problem $(\mathcal{O}_B, \mu)$: classify principally polarized abelian varieties $(A, i, \mathcal{L})$ where $A$ is an abelian variety of $\dim A = 2n$, $i: \mathcal{O}_B \into {\mathrm{End}}(A)$,and the Rosati involution has the form $*_{\mathcal{L}}: \mathcal{O}_B \rightarrow \mathcal{O}_B$, $b \mapsto \mu^{-1}\overline{b}\mu$ where $\mu \in \mathcal{O}_B$ such that $\mu^2 + \delta = 0$ and $\delta \in F^\times$.

(Shimura): The moduli functor is coarsely represented over $\mathbb{Q}$ by a complete algebraic variety $X_D/\mathbb{Q} = X_{(\mathcal{O}_B, \mu)}/\mathbb{Q}$, with $\dim X_D = n$. We havea $X_D(\mathbb{C}) = \Gamma_B\backslash \mathfrak{h}^n$, where $\mathfrak{h}$ is the Poincaré upper half plane.

Let $R_F$ be the ring of integers of $F$, $R_F \subset S \subset \mathcal{O}_B$, where $S$ is a totally real quadratic order over $R_F$. There are forgetful finite maps over $\mathbb{Q}$

$$\xymatrix{ X_{\mathcal{O}_B, \mu} \ar[r] & \mathcal{M}_S \ar[r] &... ...ert->}[r]& [A, i\vert _{R_F} , \mathcal{L}] \ar@{\vert->}[r]& [A, \mathcal{L}]}$$

where $\mathcal{M}$ is the Hilbert modular variety classifying varieties with real multiplication by the subscript. The dimensions of these moduli spaces are, respectively, $n$, $n$, $3n$, $2n^2 + n$.

The picture in $n =1$ is

$$X_{(\mathcal{O}, \mu)} \rightarrow \mathcal{M}_S \rightarrow \mathcal{A}_2 .$$

We have a tower of fields

$$\mathcal{K}_{{\mathrm{End}}(A)} = \mathbb{Q}(P) \subset \mathcal{... ...= \mathbb{Q}(P\vert _F) \subset \mathcal{K}_{A, \mathcal{L}} = \mathbb{Q}(P_0).$$

The automorphism group of $X_D = X_{(\mathcal{O}_{B},\mu)}$: is

$$(\mathbb{Z}/2)^{2r} \iso W = \frac{\mathrm{Norm}_{B_+^\times}(\Gamma_B)}{\Gamma_B \cdot F^\times}\subset {\mathrm{Aut}}_{\mathbb{Q}(X_D)}$$

where $2r = \char93 \{ \mathfrak{p}\vert\operatorname{disc}(B)\}$.

Field of definition for abelian surfaces

Let $(A, \mathcal{L})/K$ be a principally polarized abelian surface over a number field $K$ with ${\mathrm{End}}_{\overline{K}}(A) = \mathcal{O}_B$, $\operatorname{disc}(B) = D$. There is a diagram

$$\xymatrix{ & K_{{\mathrm{End}}(A)}\ar@{-}[dr]^{\text{$\vert G\ver... ...re)}} & & \mathcal{K}_{{\mathrm{End}}(A)} \\ & \mathcal{K}_{A, \mathcal{L}} & }$$

1. If $B = \left(\frac{-D, m}{\mathbb{Q}}\right)$ for $0 < m < D$, $m\vert D$, then (with Dieulefait)

   $${\mathrm{Gal}}(K_{{\mathrm{End}}(A)}/K) = \begin{cases}\mathbb{Z}... ...sqrt{D/m})$}\\ \{1\} & \text{${\mathrm{End}}_K(A) = \mathcal{O}_B$} \end{cases}$$

   
   

2. Otherwise,

   $${\mathrm{Gal}}(K_{{\mathrm{End}}(A)}/K) = \begin{cases}\mathbb{Z}... ...tensor \mathbb{Q}= \mathbb{Q}(\sqrt{-D})\\ \{1\} & \text{otherwise} \end{cases}$$

   
   

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