# Derivatives of theta functions at zero

Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions $$\theta_s(z;\tau)_k=\sum_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, \hspace{20pt} s=0,...,k-1$$ form a basis of $H^0(E, L)$. I'm especially interested in the case $k=3$ when theta functions embed the elliptic curve as Hesse cubic in $\mathbb{P}^2$.

I have the following basic question: are there formulas for derivatives of such theta functions at zero $\left.\frac{d\theta_s(z,\tau)_3}{dz}\right|_{z=0}$?

The closest result I know is Jacobi formula for Jacobi theta functions $$\theta'_{\frac{1}{2} \frac{1}{2}}=-\pi \theta_{0 0} \theta _{\frac{1}{2} 0} \theta _{0 \frac{1}{2}},$$ but (as far as I understand) it corresponds to the case $k=4$, so it does not answer my question.

• So you are asking about theta-product representation for something like$$q-2q^4+4q^{16}-5q^{25}+7q^{49}-8q^{64}+10q^{100}-11q^{121}\pm...$$ – მამუკა ჯიბლაძე Aug 26 '15 at 9:16
• (The latter seems to be $q\prod(1-q^{3(2n-1)})^2\prod(1-q^{6(2n-1)})\prod(1-q^{12n})^3$ but I don't know any reference for it) – მამუკა ჯიბლაძე Aug 26 '15 at 9:32
• I'd say I need series $\sum_{r>0} r q^{\frac{3}{2}r^2}$, this is for $s=0$, up to a constant. – Sasha Pavlov Aug 26 '15 at 15:29
• Well my $q$ is just your $q^{\frac32}$. What I wrote is for $s=1$ (for $s=2$ one just gets opposite sign). As for $s=0$, I believe you just get zero since it is the derivative of $1+\sum_{r>0}q^{r^2}(e^{6\pi irz}+e^{-6\pi irz})$ at zero. – მამუკა ჯიბლაძე Aug 26 '15 at 16:34
• Yes and btw concerning $\sum_{r>0}rq^{r^2}$ (although it seems not to be relevant to your question) there is some partial info in one of my questions here – მამუკა ჯიბლაძე Aug 26 '15 at 22:13

An eta-product identity for $k=3$, $s=1$ similar to that given in my comment above may be found in "Some eta-identities arising from theta series" by Günter Köhler (Math. Scand. 66, 1990, p. 146, identity (3)): $$\frac{\eta^2(\tau)\eta^2(4\tau)}{\eta(2\tau)}=\sum_{n=1}^\infty\left(\frac n3\right)ne^\frac{2\pi in^2\tau}3,$$ which is equivalent to $$\prod_{n=1}^\infty\frac{(1-q^n)^2(1-q^{4n})^2}{1-q^{2n}}=\sum_{n=1}^\infty\left(\frac n3\right)nq^{\frac{n^2-1}3}=1-2q+4q^5-5q^8+7q^{16}-8q^{21}+...$$ The rhs is the same as $q^{-\frac13}\sum_{n\in\mathbb Z}(3n+1)q^{\frac13{(3n+1)^2}}$, i. e., up to rescalings, $\frac\partial{\partial z}$ of your $\theta_1(z;\tau)_3$ at $z=0$.
I don't know whether such identities for other $k$ are systematically treated anywhere. Köhler also has a book "Eta Products and Theta Series Identities" but I have never looked inside. From "About this book":
(Second part) This might be a separate answer but I decided to just add it here. In "Eta-quotients and theta functions" by Robert J. Lemke Oliver (free online version available), a classification of all those theta-functions of the form $\theta_\psi(z)=\sum_n\psi(n)n^\delta q^{n^2}$ which admit an eta-product decomposition is given. Here $\psi$ is a Dirichlet character, and $\delta=0$ or $1$ according to the parity of $\psi$. There turns out to be very few of this kind - just eight in the even case and five in the odd case. The case of your $k=3$, $s=1$ is the third identity of Theorem 1.1 (2) in that paper; I am not reproducing it here as it is essentially the same as the Köhler's case above. If one allows some twists of the characters, there are a little bit more such products.