Lately I have been reading this very interesting book Mathematics++: Selected Topics Beyond the Basic Courses by Kantor, Matoušek and Šámal, especially Chapter 3 on Fourier analysis.
In Section 5: Poisson Summation Formula they pose the following questions in Exercise 5.6.
Let p be a prime and define the Gauss sum by $$ \operatorname{Gau}(r) = \sum_{x \in \mathbb{Z}_p} e(rx^2/p) \quad \text{where} \quad e(x) := \exp(2 \pi i x) $$
- Prove that $\operatorname{Gau}(rs^2) = \operatorname{Gau}(r)$ for $s \in \mathbb{Z}_p \setminus \{0\}$. (This one I have already proven since it is not that hard)
- Verify that if $-1$ is not a quadratic residue in $\mathbb{Z}_p$ then $\operatorname{Gau}(-r) = -\operatorname{Gau}(r)$.
- Let $G = \mathbb{Z}_p^2$ and define $f: G \to \mathbb{C}$ by putting $f(x_1, x_2) = e(r(x_1^2 + x_2^2)/p)$. Finally, let $H = \mathbb{Z}_p \oplus \{0\}$ be a subgroup of $G$. Apply the Poisson summation formula (defined below) to prove that $\operatorname{Gau}(1)^2 = \pm p$. (The sign depends on whether -1 is a quadratic residue.)
Poisson summation formula. Let $G$ be a finite abelian group and $H$ a subgroup of $G$. Consider $f: G \to \mathbb{C}$ and $x \in G$. Then $$ \frac{1}{|H|} \sum_{y \in H} f(x+y) = \sum_{a \in H^{\perp}} \widehat{f}(a) \chi_a(x), $$ where $\chi_a(x) := e(\sum_{i = 1}^{k}\frac{a_i x_i}{n_i})$ for $G \cong \bigoplus_{i=1}^k \mathbb{Z}_{n_i}$.
It seems to me that this is already given in the clearest manner possible, still, I am not able to deduce the 2nd and 3rd result. I have proven that $(\mathbb{Z}_p \oplus\{0\})^{\perp} = \{0\} \oplus \mathbb{Z}_p$.
Maybe someone can take it from there.