Two Equivalent Equations for the Zeros of the Jacobi Theta Function?

Question

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 $\vartheta_{a,b}$?

Answer 1

I am not sure how the Mumford claim is wrong, but simple calculations demonstrate that it is. For example, PARI/GP code:

e(x) = exp(Pi*I*x);
Thn(n,z,t,a,b) = e((n+a)^2*t) * e(2*(n+a)*(z+b));
{Th(z,t,a,b) = suminf(n=0,Thn(n,z,t,a,b)) +
               suminf(n=1,Thn(-n,z,t,a,b))};
a=.2; b=.2 ; t=1.*I;
print([Th((+a+.5)*t+b+.5,t,a,b), Th((-a+.5)*t-b+.5,t,a,b)])
[-0.766570202+4.83993375*I, 0.E-9+0.E-9*I]

It could be simple sign error which is easy to make.