Is it true that the full automorphism group of a real elliptic curve is $T\rtimes\mathbb{Z}/2\mathbb{Z}$ where $T$ is either $SO_2(\mathbb{R})$ or $SO_2(\mathbb{R})\times\mathbb{Z}/2\mathbb{Z}$? If 'yes', is there a reference?
1 Answer
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The references is Silverman's book on Arithmetic of Elliptic Curves - $Aut(E)$ depends on the $j$-invariant. It is the translation group $T$ semidirect product with $\mathbb{Z}/2\mathbb{Z}$ if $j\neq 0,1728$, and $T\ltimes \mathbb{Z}/4\mathbb{Z}$ for $j=1728$, and $T\ltimes \mathbb{Z}/6\mathbb{Z}$ for $j=0$. This is valid for all fields of characteristic not $2$ or $3$.
Actually, in Silverman $Aut(E)$ of an elliptic curve $E$ is defined so that it is a finite group, namely $\mathbb{Z}/2\mathbb{Z}$ if $j\neq 0,1728$, and and $\mathbb{Z}/4\mathbb{Z}$ for $j=1728$, and $\mathbb{Z}/6\mathbb{Z}$ for $j=0$.
answered Dec 10, 2015 at 15:39
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$\begingroup$ Do there exist automorphisms of order, for example, 6 (preserving 0) defined over $\mathbb{R}$? $\endgroup$– NikoDec 10, 2015 at 15:46
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$\begingroup$ $Aut(E)=\mathbb{Z}/6\mathbb{Z}$ only for elliptic curves with $j$-invariant zero. $\endgroup$ Dec 10, 2015 at 15:48