Statement

Suppose that

• C is a compact Riemann surface
• g is the genus of C
• \theta is the Riemann theta function of C, a function from \mathbb{C}^g to \mathbb{C}
• E is a prime form on C\times C
• u, v, x, y are points of C
• z is an element of \mathbb{C}^g
• \omega is a 1-form on C with values in \mathbb{C}^g

The Fay's identity states that

\begin{align} &E(x,v)E(u,y)\theta\left(z+\int_u^x\omega\right)\theta\left(z+\int_v^y\omega\right)\

&E(x,u)E(v,y)\theta\left(z+\int_v^x\omega\right)\theta\left(z+\int_u^y\omega\right)\

&E(x,y)E(u,v)\theta(z)\theta\left(z+\int_{u+v}^{x+y}\omega\right) \end{align}

with

\begin{align} &\int_{u+v}^{x+y}\omega=\int_u^x\omega+\int_v^y\omega=\int_u^y\omega+\int_v^x\omega \end{align}

References