# Endomorphisms of an elliptic curve

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.

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 $$\alpha_0$$.

• 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? – uhhhhidk Jul 26 at 20:44
• 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 $a\Lambda \subset \Lambda$ – uhhhhidk Jul 26 at 23:11