Jacobi forms and Kato's modular units

Question

this is pretty much just a silly literature question; apologies in advance. Kato uses the following theta function (or slight variants thereof) in his construction of his Euler system: $$\Theta(\tau, z) = q^{1/12}(e^{\pi iz} - e^{-\pi iz}) \prod_{n\ge 1} (1-q^n e^{2\pi iz})(1-q^ne^{-2\pi i z})$$ where $q=e^{2\pi i \tau}$. This is the theta function I like as someone who leans heavily towards the algebraic side of things; it's essentially the de Rham realization of a polylogarithm class in the $H^{1,1}$ motivic cohomology of some universal elliptic curve. However, I can't quite figure out the relation to some classical formulas for the Jacobi theta function. It's been stated in several places (for example, Scholl's notes) that the above theta function is essentially the same as the classical "half-integral weight Jacobi form," which I guess is the series $$ \vartheta(\tau, z)=\sum_{n\in \mathbb{Z}} e^{\pi i n^2 \tau + 2\pi i n z}= \sum_{n\in \mathbb{Z}} q^{n^2/2} e^{2\pi i n z}$$ which has the similar-looking product expansion $$\vartheta(\tau, z)= \prod_{n=1}^\infty (1-q^n)(1-e^{2\pi i z}q^n)(1-e^{-2\pi iz}q^n)$$ (In particular, Scholl says it essentially coincides with the Jacobi theta series $\vartheta_1$, which didn't prove so helpful since I've found that naming conventions for these several classical Jacobi forms are really confusing and inconsistent in the literature. But all of them have very similar forms and are related by a few simple transformations, from what I can tell. It seems they're basically the orbit of the Jacobi form I wrote down under translation by the half-integral lattice, I gather this has something to do with transformations under the action of the metaplectic cover but don't fully understand the significance.)

These almost look same, but I can't quite figure how to get from one to the other. The denominator in the leading power of $q$ bothers me since it implies a monodromy obstruction in Kato's case. Most notably, the latter function is a triple product whereas the former isn't; the Jacobi triple product identity is my understanding of how to expand the classical Jacobi-type products, so the fact that Kato's theta function is missing one of the "triple product" ingredients throws me off, and I can't work out how to relate them.

Is this discussed somewhere in the literature? I'd really love to be able to clarify the relationship here.

Answer 1

The triple product in the Jacobi theta function $\vartheta(\tau,z)$ can be rewritten \begin{equation*} \prod_{n \geq 1} (1-q^n) \prod_{n \geq 0} (1-q^n e^{2\pi i(z+\frac{\tau}{2}+\frac12)}) \prod_{n \geq 1} (1-q^n e^{-2\pi i(z+\frac{\tau}{2}+\frac12)}) \end{equation*} ($2\pi i n$ should be replaced by $2\pi iz$ in your equation).

On the other hand, the theta function used by Kato is essentially the Weierstrass $\sigma$ function (see for example Silverman, Advanced topics in the arithmetic of elliptic curves, I.6.4). This is not surprising as the $\sigma$ function can be used to construct elliptic functions with prescribed divisors (op. cit., I.5.5).

If you work things out, you will see that the discrepancy is essentially the cube of the infinite product $\prod_{n \geq 1} (1-q^n)$, which brings into play $\Delta^{1/8}$, explaining the factor $q^{1/8}$.

Note that Kato really uses the function $\Theta(\tau,z)^{c^2} \Theta(\tau,cz)^{-1}$, which is associated to the divisor $c^2(0) - \sum_{x \in E[c]} (x)$ on the universal elliptic curve. Since this divisor has degree 0, the term $\prod (1-q^n)$ will cancel out.