  In this paper we obtain a new sum-product estimate in prime fields. In
particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le
p^{64/117}$ then $$ \max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}. $$ Our argument
builds on and improves some recent results of Shakan and Shkredov which use the
eigenvalue method to reduce to estimating a fourth moment energy and the
additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty
comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound
of Rudnev rather than a point line incidence bound of Stevens and de Zeeuw as
done by Shakan and Shkredov.
