$\mathrm{USp}(4)$ 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 $n\in\mathbb{Z}$.