Simple automorphism proof with lattice elliptic curves?

Question

For a lattice $\Lambda_1=\mathbb{Z}+\mathbb{Z}i$, find Aut($E_1$) where $E_1(\mathbb{C})=\mathbb{C}/\Lambda_1$.

So I know that End($E$) $ =\{\beta \in \mathbb{C} $| $\beta\Lambda \subseteq \Lambda\}$ and Aut($E$) $=\{\beta \in \mathbb{C} $| $\beta\Lambda =\Lambda\}$ but I'm really not sure how to use this information to find the correct automorphisms.

What's a good way to find the Aut($E_1$)?

Answer 1

Since you know that ${\rm End}(\Bbb C/\Lambda)=\{\lambda\in\Bbb C\mid\lambda\Lambda\subseteq\Lambda\}$ it should be clear that $$ {\rm Aut}(\Bbb C/\Lambda)=\{\lambda\in\Bbb C\mid\lambda\Lambda=\Lambda\}. $$ When $\Lambda=\Bbb Z\otimes\Bbb Zi=\Bbb Z[i]$ the latter set is identified with the set of invertible elements in the ring $\Bbb Z[i]$ i.e. $$ \{1,-1,i,-i\}. $$

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A variant is to observe that for any $\lambda\in\Bbb Z[i]$ the set $\lambda\Lambda=\lambda\Bbb Z[i]$ is an ideal with $$ \left|\frac\Lambda{\lambda\Lambda}\right|=|\lambda| \qquad\text{(complex norm)} $$ (this can be understood observing that multiplying by $z\in\Bbb C$ an area of a region $A\subset\Bbb C=\Bbb R^2$ gets multiplied by a factor $|z|$). Thus $\lambda\in{\rm Aut}(\Bbb C/\Lambda)$ when $|\lambda|=1$, leading to the same result.

Answer 2

First, I would suggest determining $End(E_1)$. To do this, you might note that if $\alpha\in\mathbb{C}$ and $\alpha(\mathbb{Z}+i\mathbb{Z})\subset\mathbb{Z}+i\mathbb{Z}$ if and only if $\alpha\in\mathbb{Z}+i\mathbb{Z}$ and hence $End(E_1) = \mathbb{Z}+i\mathbb{Z}$. Then, the automorphisms will be the endomorphisms of norm 1, that is the units of $\mathbb{Z}[i]$, the cyclic group generated by $i$, which as a set consists of the elements $\{1,-1,i,-i\}$.