If $( A B | C D)$ is a symplectic matrix with entries in the finite field with two elements, is it necessarily the case that $\sum_{i,j,k} a_{ij}b_{ij}c_{ik}d_{ik} = 0$?

This arose in connection with some calculations involving theta functions. There seems to be some indication that it might be true, and I could not come up with a counterexample; but experts in the area may well know the answer right off.

Thanks for any help.

`$(AB|CD)=\begin{pmatrix}A&B\\C&D\end{pmatrix}$`

is a block matrix over`$\mathbb F_2$`

. Your notation suggests moreover that $A,B,C,D$ are all squares. "symplectic" presumably means that it fixes (infinitesimally) some particular symplectic form, which (if I had to guess) you are taking to be`$(AB|CD)=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$`

, where by "$1$" I mean the identity square matrix of whatever size you're working with. If this is all correct, fine, but you should edit the question to make precise the notation. – Theo Johnson-Freyd Oct 1 '10 at 3:45