# Properties

 Label 1.4.A.1.1a Name $$\mathrm{USp}(4)$$ Weight $1$ Degree $4$ Real dimension $10$ Components $1$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{USp}(4)$$ Component group $$C_1$$

$\mathrm{USp}(4)$ is the Sato-Tate group of a generic abelian surface over a number field.

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $10$ Components: $1$ Contained in: $\mathrm{USp}(4)$ Rational: yes

## Identity component

 Name: $\mathrm{USp}(4)$ $\mathbb{R}$-dimension: $10$ Description: $\Bigl\{ A\in \mathrm{GL}_4(\mathbb{C}):A^{-1}=\bar{A^{\mathrm{t}}},\ A^{\mathrm{t}}\Omega A = \Omega\Bigr\}$ Symplectic form: $\Omega:=\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,u,\bar u,\bar u)$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $3$ $0$ $14$ $0$ $84$ $0$ $594$ $0$ $4719$
$a_2$ $1$ $1$ $2$ $4$ $10$ $27$ $82$ $268$ $940$ $3476$ $13448$ $53968$ $223412$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1$ $1$ $2$ $2$ $3$ $4$ $5$ $8$ $14$ $10$ $14$ $24$ $44$ $84$ $27$ $43$ $78$ $149$ $294$ $594$ $82$ $142$ $270$ $534$ $1084$ $2244$ $4719$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0&0\\0&0&0&1&0&0&0&0&0&0\\0&0&0&0&1&0&0&0&0&0\\0&0&0&0&0&1&0&0&0&0\\0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&0&0&1\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&1&1&1&1&1&1&1&1\end{bmatrix}$

## Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.