Elliptic curve vs. Elliptic function

Question

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.

Answer 1

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 $\mathbb{C}[x,y]/(y^2 - 4x^3 - g_2(\Lambda)x - g_3(\Lambda))$.

Answer 2

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 $f$.