I'm trying to find the 0's for the Jacobi theta function with characteristic:
$$
\vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b \in \mathbb{R}.
$$
In Mumfords Lectures on Theta, the 0's are given at the points:
$$(a + p +1/2)\tau + (b + q + 1/2) $$ where $p,q$ are integers.
OOH, a previous question on Stackexchange:
the zeros of theta function? gives the zeroes as being at: $$z=\!(p-\!\tfrac 12\!-\!a)\tau+\!\tfrac 12\!-b-q$$
Unfortunately, I can't follow the proof well enough in the latter to check for a mistake, but from what I can tell the latter actually gives the 0's of $\vartheta_{-a,-b}$. Which of these equations gives the correct 0's of $\vartheta_{a,b}$?
Best Answer
I am not sure how the Mumford claim is wrong, but simple calculations demonstrate that it is. For example, PARI/GP code:
It could be simple sign error which is easy to make.