  Sharp restriction theory and the finite field extension problem have both
received a great deal of attention in the last two decades, but so far they
have not intersected. In this paper, we initiate the study of sharp restriction
theory on finite fields. We prove that constant functions maximize the Fourier
extension inequality from the parabola $\mathbb{P}^1\subset
\mathbb{F}^{2\ast}_q$ and the paraboloid $\mathbb{P}^2\subset
\mathbb{F}_q^{3\ast}$ at the euclidean Stein-Tomas endpoint; here,
$\mathbb{F}_q^{d\ast}$ denotes the (dual) $d$-dimensional vector space over the
finite field $\mathbb F_q$ with $q=p^n$ elements, where $p$ is a prime number
greater than $3$ or $2$, respectively. We fully characterize the maximizers for
the $L^2\to L^4$ extension inequality from $\mathbb{P}^2$ whenever $q\equiv
1(\text{mod}\, 4)$. Our methods lead to analogous results on the hyperbolic
paraboloid, whose corresponding euclidean problem remains open. We further
establish that constants maximize the $L^2\to L^4$ extension inequality from
the cone $\Gamma^3:=\{(\boldsymbol{\xi},\tau, \sigma)\in \mathbb{F}^{4\ast}_q:
\tau\sigma=\boldsymbol{\xi}^2\}\setminus \{{\bf 0}\}$ whenever $q\equiv
3(\text{mod}\, 4)$. By contrast, we prove that constant functions fail to be
critical points for the corresponding inequality on $\Gamma^3\cup \{{\bf 0}\}$
over $\mathbb{F}_p^4$. While some inspiration is drawn from the euclidean
setting, entirely new phenomena emerge which are related to the underlying
arithmetic and discrete structures.
