# Elliptic curve vs. Elliptic function

I am a bit unsure about the relation between elliptic curves and elliptic functions.

I believe that there is a one to one correspondence between elliptic curves and Weierstrass's elliptic functions (via a differential equation), which in turn are in a one to one correspondence with complex lattices. Is that correct?

For general elliptic functions, is there a similar differential equation? And then a corresponding variety with a group law (from the underlying lattice, like for elliptic curves)? I guess not, but perhaps I am missing something.

I would like a second opinion from someone more experienced than me in this area. Many thanks in advance.

• Short answer is yes: see chapter 2 of Hirzebruch and Jung's "Manifolds and Modular Forms." A Weierstrass function can be used to parameterize an elliptic curve, and an elliptic curve determines a $p$-function. These processes are inverse to one another (maybe excluding some degenerate cases). Jul 29, 2020 at 21:53
• @ElliotG Thank you, I wanted a short answer. Do you know something about the second part of the question? I guess the short answer to the question is no, but the difficulty is that textbooks usually do not mention ideas which do not work.
– Loic
Jul 30, 2020 at 8:33

You are not quite correct in your statement "there is a 1-to-1 correspondence between Weierstrass' elliptic functions and elliptic curves $$E/\mathbb{C}$$".

The true statement is that there is a 1-to-1 correspondence between elliptic curves $$E/\mathbb{C}$$ and lattices $$\Lambda \subset \mathbb{C}$$ up to homothety. The parametrisation is given by $$\wp$$ (this statement is Silverman AEC Corollary VI.5.1.1 and Prop VI.5.2(b) - and is even more precise in Theorem VI.5.3). Moreover for a given $$\Lambda$$ we construct an $$E$$ as $$E : y^2 = 4x^3 + g_2(\Lambda)x + g_3(\Lambda)$$ and $$E(\mathbb{C}) \cong \mathbb{C}/\Lambda$$ is a complex analytic group isomorphism via $$z \mapsto [\wp(z): \wp'(z): 1]$$.

Elliptic functions on $$\Lambda$$ form a field denoted $$\mathbb{C}(\Lambda) = \mathbb{C}(\wp, \wp')$$ (the equality is Silverman VI, Theorem 3.2) and under the isomorphism above we get an isomorphism $$\mathbb{C}(\Lambda) \cong \mathbb{C}(E)$$ the function field of $$E$$ (i.e., the fraction field of $$\mathbb{C}[x,y]/(y^2 - 4x^3 - g_2(\Lambda)x - g_3(\Lambda))$$.

This is probably only a partial answer but it's too long for a comment.

The relation between the Weierstrass elliptic function $$\wp(z):=\wp(z, \omega_1, \omega_3)$$ with fundamental periods $$2 \omega_1$$ and $$2 \omega_3$$ (that is the one associated with the lattice $$\Lambda:=\langle 2\omega_1, 2\omega_3 \rangle$$) is indeed the differential equation $$(\wp'(z))^2 = 4\wp^3(z) − g_2\wp(z) − g_3$$ where $$g_2=60G_4$$ and $$g_3 = 140G_6$$ are defined using the Eisenstein series $$G_{2n}$$. This implies that the pair $$(\wp,\wp')$$ parametrizes an elliptic curve.

Now if $$f$$ is any elliptic function with $$2 \omega_1$$ and $$2\omega_3$$ as fundamental periods, we can find rational functions $$R_1$$ and $$R_2$$ of one complex variables such that $$f(z) = R_1(\wp(z)) + R_2(\wp(z))\wp'(z)$$ This in turn should produce a differential equation for $$f$$. This should (I am not an expert on elliptic curves) parametrize an elliptic curve associated with $$f$$.

• A partial answer, indeed, but valuable nonetheless. The first part makes my question more explicit (I was too lazy to do so, thank you for that). The second provides additional information, although it is not directly related to the question.
– Loic
Jul 30, 2020 at 8:38