The "$\mathbb{Z}_n$-theta function" - what is it? Is it being studied somewhere?

Question

The Jacobi theta function is well known: $$\theta(z, \tau) = \sum_{n=-\infty}^\infty \mathrm{e}^{\pi i n^2 \tau + 2\pi i nz}$$ In Shahn Majid's "Foundations of Quantum Group Theory", you'll find a sort of finite theta function in example 2.1.11 on page 45, which he calls "$\mathbb{Z}_n$-theta function": $$\theta_d(a) = \frac{1}{d}\sum_{n=0}^d \mathrm{e}^{\frac{2\pi i}{d} (n^2 + an)} \qquad d \in \mathbb{N}$$ I had never heard of such a function before. Google doesn't give anything promising. It sort of looks like a theta function with $\tau = \frac{2}{d}$ and $z = \frac{a}{d}$, but the sum is finite (like a kind of regularisation?). Is this studied anywhere? Can it be derived from the usual theta function? Are there known identities involving this?

Answer 1

Often we have an object and we desire to find analogues of it in other settings. Sometimes the definition of the object does not directly translate from the original setting to the new setting. The solution is to rewrite the definition in a way that it can be translated directly into the new setting.

---

Consider the Fourier transform. This I think is prudent to highlight because it is related to the theta function - indeed $\theta(z,\tau)$ appears to be a Fourier series with a "quadratic augmentation."

Originally, if $f(x)$ is a nice function we have $\hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i \xi x}dx$. How do we translate this into a setting that doesn't necessarily have real or complex numbers? Obviously we can't just write down the expression $e^{-2\pi ix\xi}$. Notice, for each $\xi\in\Bbb R$, the function $\chi_\xi(x):=e^{-2\pi i\xi x}$ is a continuous group homomorphism $\Bbb R\to S^1$, where $S^1$ is the group of complex numbers which have absolute value $1$ and multiplication (and the subspace topology inherited from $\Bbb C$). This is a considerably general type of object, a character.

For $G$ a sufficiently nice (locally compact abelian) group, the set $\widehat{G}$ of characters $G\to S^1$ equipped with pointwise multiplication becomes a group, plus $G$ also admits its own (Haar) measure, which allows us to integrate over it. Then for functions $f:G\to\Bbb C$, we can define the Fourier transform as a function $\hat{f}:\widehat{G}\to\Bbb C$ given by $\hat{f}(\chi):=\int_G f(g)\chi(g)d\mu(g)$ (for $\chi\in\widehat{G}$).

In particular, we can equip the finite cyclic group $G=\Bbb Z/n\Bbb Z$ with the discrete topology and talk about the discrete Fourier transform. The characters $\Bbb Z/n\Bbb Z\to S^1$ are given by $k+n\Bbb Z\mapsto e^{2\pi i ak/n}$ for various $a\in\Bbb Z/n\Bbb Z$, and so with $f:\Bbb Z/n\Bbb Z\to\Bbb C$ we have $\hat{f}(a)=\frac{1}{n}\sum_{k=0}^n f(k)e^{2\pi i ak/n}$.

---

The book says the $\Bbb Z/n\Bbb Z$-theta function is the "$\Bbb Z/n\Bbb Z$-Fourier transform of a Gaussian." What does it mean by "a Gaussian"? The usual "Gaussian functions" look like $\alpha e^{\beta x^2}$ for some choice of parameters $\alpha,\beta$. With complex numbers this is the same as $\alpha e^{2\pi i \beta x^2}$. One might guess then that with a ring $R$ whose underlying additive group is nice and topological, if $\chi:(R,+)\to\Bbb C^\times$ is a group homomorphism then $\chi(x^2)$ is a Gaussian function. In the setting of $\Bbb Z/n\Bbb Z$, the prototypical Gaussian function will be $\chi:k+n\Bbb Z\mapsto\exp(2\pi i k^2/n)$, and the $\Bbb Z/n\Bbb Z$-Fourier transform of this will yield exactly this so-called $\Bbb Z/n\Bbb Z$-theta function.

This analogy is imperfect. The original theta function is given by a Fourier series, which is sort of but not exactly a Fourier transform for the group $\Bbb Z$. But then the "Gaussian" is allowed to have a complex parameter, so in a sense the theta function is a bit more prettified and prepped for things like heat equations and modularity. However I think the preceding paragraph is what the author is thinking, especially due to the phrase "Fourier transform of a Gaussian."

I am not sure this usage of "Gaussian" is ubiquitous. For example, in the context of Tate's thesis in number theory, Gaussians are the fixed points of (additive) Fourier transforms, and then the multiplicative Fourier transform of the Gaussians are the Gamma factors. At any rate, there is actually quite a lot of literature on these discrete theta functions, only they go by a different name: you will want to search for generalized quadratic Gauss sums. One of my favorite expositions on Gauss sums that I've come across is this summer journal.