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.