## Background

Fix an odd prime $p$ and set $\zeta=e^{2\pi i/p}$. We define the *quadratic Gauss sum* as

$$g=\sum_{n=0}^{p-1} \zeta^{n^2}.$$

It's pretty easy to show that

$$g^2= \begin{cases} p & \textrm{if } p\equiv 1 \mod 4 \\\ -p & \textrm{if } p\equiv 3 \mod 4, \end{cases}$$

and from this we can deduce quadratic reciprocity; it's harder to determine the modulus. We can actually find an explicit formula for $g$, namely:

$$g= \begin{cases} \sqrt{p} & \textrm{if } p\equiv 1 \mod 4 \\\ i\sqrt{p} & \textrm{if } p\equiv 3 \mod 4. \end{cases}$$

This is the result I refer to for the remainder of the question.

## Question

Can we categorify this result?

By *categorification*, I mean the opposite of *decategorification*, and by decategorification, I mean the process of removing structure by e.g. taking the cardinality of a set or the dimension of a vector space. (Thus an example of categorification would be interpreting some combinatorial identity of positive integers as a bijection between sets.) This is intentionally vague, because there are plenty of people who have a much better idea of what constitutes categorification than I do, so feel free to interpret "categorification" liberally.

## Motivation

Gauss's original proof of our result uses q-binomial coefficients. (A modern exposition of this proof can be found in "The determination of Gauss sums" by Bruce C. Berndt and Ronald J. Evans.)

Now, $q$-binomial coefficients can be categorified by Grassmannian varieties. What I mean by that is: the $q$-binomial coefficient $\binom{n}{k}_q$ is the number of $k$-dimensional subspaces of an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, i.e. the cardinality of the Grassmannian $\textrm{Gr}(n,k)$. Basically, I'm wondering if there is some way this can be connected to the formula for the quadratic Gauss sum, seeing as how the formula is clearly related to the properties of $q$-binomial coefficients.

whatto prove about the objects yet! $\endgroup$