How to construct the function on torus with two distinct simple poles?

Question

Consider torus $T=C/\Lambda$ where $\Lambda$ is a rank 2 lattice and $C$ is complex plane. Then $g(T)=1$. From riemann roch, I deduce that $P\neq Q, h^0(P+Q)=2+(1-1)+0=2$ where $0=h^0(K_T-P-Q)$ by $deg<0$ and $K_T$ is the canonical divisor. Since $h^0(P)=1$ and it contains only constant functions, there is a meromorphic function with simple pole at $P$ and $Q$.

For $P=Q$, the construction of such function is wierstrass P function.

I tried Mittag Leffler construction. However due to correction of a holomorphic polynomial for compact convergence, I cannot construct such function from simple poles by scratch.(i.e. Considering $\sum_{\lambda\in\Lambda}(\frac{1}{z-P+\lambda}-f_\lambda(z))+\sum_{\lambda\in\Lambda} (\frac{1}{z-Q+\lambda}+g_\lambda(z))$ where $f,g$ are polynomials s.t I can have compact convergence. However this clearly breaks translation symmetry in general.)

$\textbf{Q:}$What about $P\neq Q$? I saw there is a function meromorphic at both $P$ and $Q$ with simple pole. However, none of the books talk about it.

Answer 1

I'll just sketch the ideas:

Define Jacobi's theta function for $\tau \in \mathbb{H}, z \in \mathbb{C}$ $$\vartheta_\tau(z) := \sum_{n \in \mathbb{Z}} e^{\pi i (n^2 \tau + 2nz)}.$$ This defines a holomorphic function which satisfies $\vartheta_\tau(z + 1) = \vartheta_\tau(z)$ and $\vartheta_\tau(z + \tau) = e^{-\pi i (\tau + 2z)} \vartheta_\tau(z)$. Moreover, $\vartheta_\tau(z)$ has its only zero of order $1$ (up to translation by $m + n \tau$, $m,n \in \mathbb{Z}$ at $z = (1 + \tau)/2$. We now define $$\vartheta_\tau^{(x)} = \vartheta_\tau(z + (1 + \tau)/2 - x)).$$ This function has its only zeros at $x + L$, $L = \mathbb{Z} \oplus \tau \mathbb{Z}$. Now, quotients $$R(z) = \frac{\prod_{i = 1}^m \vartheta_\tau^{(x_i)}(z)}{\prod_{j = 1}^n \vartheta_\tau^{(y_j)}(z)}$$ define $L$-periodic meromorphic functions (i.e. functions on the torus $\mathbb{C} / L$) if and only if $m = n$ and $\sum_{i = 1}^m x_i - \sum_{j = 1}^n y_j = 0$. The poles and zeros of $R(z)$ are obvious.

Edit: Bonus Exercise: The functions $R(x)$ are up to constant factors all meromorphic functions of the torus $\mathbb{C} / L$.

Answer 2

Let $\,f(z) := \sigma(z-z_1)\,\sigma(z-z_2)/ (\sigma(z-p_1)\,\sigma(z-p_2)) \,$ where $\,\sigma\,$ is the Weierstrass sigma function and where $\,z_1+z_2 = p_1+p_2.\,$ The doubly periodic function $\,f(z)\,$ has zeros at $\,z_1,z_2\,$ and poles at $\,p_1,p_2$ using the lattice $\;\Lambda.\,$ This generalizes to any finite number $\,n\,$ of zeros and poles. This is a known result. For example, see Harris Hancock, Lectures on the Theory of Elliptic Functions, page 439.