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If I am correct, an endomorphism is a homomorphism from one group to itself. In the case of an elliptic curve, we'd need a map $\phi: E \rightarrow E$. Ive seen in some places that you can get such a map by defining $\phi_n(P)=[n]P=P+P...+P$ (n times). This means if we represent P as $(\wp(t),\wp'(t))$, (over some arbitrary lattice) $\phi_n((\wp(t),\wp'(t)))=(\wp(nt),\wp'(nt))$.

Using this definition of a endomorphism found that $\phi_i$ is also a valid map if $E$ is parametrized by $(\wp(t,[1,i]),\wp'(t,[1,i]))$ since $\phi_i((\wp(t,[1,i]),\wp'(t,[1,i])))=(\wp(it,[1,i]),\wp'(it,[1,i]))=(-\wp(t,[1,i]),i\wp'(t,[1,i]))$.

That case was simple because the lattice $[1,i]$ is unchanged when divided by $i$. In a more general case where $E$ is parametrized by $(\wp(t,\Lambda),\wp'(t,\Lambda))$, how would one find other endomorphisms of E? Also, how does one prove these are homomorphisms? I havent been able to find any sort of reasoning for using such a map and think it would be useful to actually see why this is a homomorphism. Thanks for any help.

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You are asking for all the holomorphic (because they are algebraic) endomorphisms of $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice.

Note that they correspond to holomorphic endomorphisms $f:\mathbb{C} \rightarrow \mathbb{C}/\Lambda$.

They lift to the universal cover into $g: \mathbb{C} \rightarrow \mathbb{C}$ with $g(0)=0$. Thus $(x,y) \rightarrow g(x+y)-g(x)-g(y)$ is continuous and $\Lambda$-valued, therefore $g(z)=\alpha z$ for some $\lambda \in \Lambda$.

Since $\Lambda \subset \ker(g)$, $\alpha \Lambda \subset \Lambda$.

Assume $f$ is not trivial: thus $\alpha$ is not an integer.

$x \in \Lambda \longmapsto \alpha x \in \Lambda$ can be seen as an endomorphism of $\mathbb{Z}^2$. Let $\chi$ be its characteristic polynomial: by “Cayley-Hamilton”, $\chi(\alpha)=0$, thus $\alpha$ is a quadratic integer, and $\Lambda=e_1(\mathbb{Z}1\oplus\mathbb{Z}\alpha)$.

Thus, for most lattices $\Lambda$, (each one but countably many, up to similarity), the only endomorphisms of $\mathbb{C}/\Lambda$ are integer multiples of identity.

On the other lattices, there are some “exotic” multiplications allowed (they are the “complex multiplication” or CM elliptic curves/lattices), but all endomorphisms are multiplications by some complex number $\alpha$ such that $\alpha\Lambda \subset \Lambda$. The set of such $\alpha$ is a $\mathbb{Z}[\alpha_0]$ for some quadratic integer $\alpha_0$.

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  • $\begingroup$ Is there any way to think about this in terms of the weistrass function? Why is $\wp(\alpha z,\Lambda)$ special when $\alpha \Lambda \subset \Lambda$? I believe I saw somewhere that $\wp(\alpha z,\Lambda)$ is a rational function in $\wp(z,\Lambda)$ if $\alpha \Lambda \subset \Lambda$, is this related to the fact to the endomorphisms of E or is it just a coincidence? $\endgroup$ – uhhhhidk Jul 26 at 20:44
  • $\begingroup$ I think I understand now: I forgot the condition that $\phi_a (O)=O$. The only $a$ that satisfy this property are $a$ such that $a\Lambda \subset \Lambda$ because $(\wp (a\lambda), \wp'(a\lambda))$ ($\lambda \in \Lambda$) wont be equal to the identity unless $a\Lambda \subset \Lambda$ $\endgroup$ – uhhhhidk Jul 26 at 23:11

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