# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
### 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 ...
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 ...