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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.

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  • $\begingroup$ So you are asking about theta-product representation for something like$$q-2q^4+4q^{16}-5q^{25}+7q^{49}-8q^{64}+10q^{100}-11q^{121}\pm...$$ $\endgroup$ – მამუკა ჯიბლაძე Aug 26 '15 at 9:16
  • $\begingroup$ (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) $\endgroup$ – მამუკა ჯიბლაძე Aug 26 '15 at 9:32
  • $\begingroup$ 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. $\endgroup$ – Sasha Pavlov Aug 26 '15 at 15:29
  • $\begingroup$ 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. $\endgroup$ – მამუკა ჯიბლაძე Aug 26 '15 at 16:34
  • $\begingroup$ 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 $\endgroup$ – მამუკა ჯიბლაძე Aug 26 '15 at 22:13
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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":

The author brings to the public the large number of identities that have been discovered over the past 20 years, the majority of which have not been published elsewhere.

(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.

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