0
votes
0answers
40 views

References for the conformal equivalence of the space of complex 1-tori and C?

What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
16
votes
1answer
349 views

The importance of modular forms

I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...
4
votes
0answers
133 views

Weierstrass $\wp$-Function Addition Property

Consider the function $$ \det\left( \begin{array}{ccccc} &1 &\wp(z) &\wp'(z) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(-z-w) &\wp'(-z-w) \end{array} \right)=f(z) $$ I'm ...
0
votes
0answers
101 views

Why is Every Elliptic Function of Order $2$ the Möbius Tranform of a $\wp$-function?

I'm trying to prove that every elliptic function of order $2$ has the form $$f(z)=\frac{a\wp(z-z_0)+b}{c\wp(z-z_0)+d}$$ I've got the following so far. Let $f$ be an elliptic function of order 2. ...
3
votes
0answers
145 views

Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent. Definition 1 $\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
4
votes
1answer
787 views

A Torus and the Weierstrass P function?

Let $\wp$ be the Weierstrass function. From what I understand, $\wp$ maps the torus to $CP^1 \times CP^1$ in the following way: $a \mapsto (\wp(a),\wp'(a)) = (z,w)$ Furthermore, the image of this ...