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 \(C_1\)

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$\mathrm{USp}(4)$ 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

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

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 $n\in\mathbb{Z}$.