Properties

Label 1.2.3.1.1a
  
Name \(\mathrm{SU}(2)\)
Weight 1
Degree 2
Real dimension 3
Components 1
Contained in \(\mathrm{USp}(2)\)
Identity Component \(\mathrm{SU}(2)\)
Component group \(C_1\)

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$\mathrm{SU}(2)$ is the Sato-Tate group of any non-CM elliptic curve $E$ over a number field $K$.

As conjectured by Mikio Sato and John Tate and proved by Richard Taylor et al., when $K=\Q$ the normalized Frobenius traces of $E$ are equidistributed with respect to the Haar measure on $\mathrm{SU}(2)$.

Invariants

Weight:$1$
Degree:$2$
$\mathbb{R}$-dimension:$3$
Components:$1$
Contained in:$\mathrm{USp}(2)$
Rational:$\mathrm{True}$

Identity Component

Name:$\mathrm{SU}(2)$
Index:$1$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{bmatrix}:\alpha\bar\alpha+\beta\bar\beta = 1,\ \alpha,\beta\in\mathbb{C}\right\}$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $2$ $0$ $5$ $0$ $14$ $0$ $42$ $0$ $132$

Additional information

The Sato-Tate conjecture for elliptic curves over number fields has been proved for totally real fields (including $\Q$) [10.4007/annals.2010.171.779, 10.1007/s10240-008-0015-2], and CM fields [arXiv:1812.09999].