The prime form of a Riemann surface of genus $g$ can be defined in terms of multi-dimensional $\vartheta$-functions as follows:

$$E(z,w) = \frac{\vartheta\Big[\genfrac{}{}{0pt}{}{\vec{a}}{\vec{b}} \Big](\int_w^z \vec{\omega}|\Omega)}{\zeta(z)\zeta(w)}.$$

Here, $\vec{\omega}$ is the vector of holomorphic differentials $\omega_j (j=1,\cdots,g)$ and $\Omega$ is the $g\times g$ period matrix. Also

$$\zeta(z) = \sum_{j=1}^g \omega_j \frac{\partial}{\partial x_j} \vartheta\Big[\genfrac{}{}{0pt}{}{\vec{a}}{\vec{b}} \Big](0|\Omega).$$

$a,b$ are $g$-dimensional vectors such that $a_j,b_j$ take values $0,1/2$. It is also required that $4 \vec{a}\cdot\vec{b}$ is an odd-integer so that $$E(z,w) = - E(w,z),$$ which follows from the definition of the $\vartheta$-function with characteristics

$$\vartheta\Big[\genfrac{}{}{0pt}{}{\vec{a}}{\vec{b}} \Big](\vec{z}|\Omega) = \sum_{\vec{n}\in \mathbb{Z^g}} e^{i \pi (\vec{n} + \vec{a})\cdot \Omega \cdot (\vec{n} + \vec{a}) + 2\pi i(\vec{n}+\vec{a})\cdot (\vec{z}+\vec{b})}$$.

My question is the following: how does $E(z,w)$ transform around the $A$ and $B$ cycles? I've looked in various books (Fay's book, the Tata lectures) and I've found that the answers only agree up to a sign. So after parallel transport around an $A_j$-cycle

$$E(z,w) \to \pm E(z,w)$$

and around a $B_j$-cycle

$$E(z,w) \to \pm E(z,w) e^{-i\pi \Omega_{jj} - 2 \pi i \int_w^z \omega_j}$$.

From looking at an example on the torus ($g=1$), it appears to me that correct answer is that $E$ picks up a negative sign around both cycles since its dependence on $z$ comes through the numerator. Could someone help me fix this?

EDIT: I've also seen that under a modular transformation,

$$E(z,w) \to E(z,w) \exp(\int_z^w\vec{\omega}(C \Omega + D)^{-1} C\int_z^w \vec{\omega})$$,

which I'm also having trouble showing based on the definition in terms of $\vartheta$-functions. Any help with this also appreciated!



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