1
vote
2answers
73 views

Constructing a meromorphic function

I need help with the following problem. "Let $C : y^2 = x^3 − 5x^2 + 6x$ be a cubic curve with the standard group law. Find a meromorphic function on $C$ having the pole of order two at ...
18
votes
1answer
398 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
141 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
109 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
152 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
843 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 ...