A complete characterization of a family of permutation trinomials over ${\mathbb{F}}_{p^2}$

Let $p>3$ be a prime and let $$f_{{\lambda }_1,{\lambda }_2}(x)=x^{p^2-p+1}+{\lambda }_1{x}^{p^2}+{\lambda }_2{x}^{2p-1}\in {\mathbb{F}}_{p^2}[x]\mathrm{.}$$ We determine all pairs $({\lambda }_1,{\lambda }_2)\in ({\mathbb{F}}_{p^2}{)}^2$ for which $f_{{\lambda }_1,{\lambda }_2}$ is a permutation polynomial of ${\mathbb{F}}_{p^2}$. The final classification consists of three explicit families. The first one is the binomial case ${\lambda }_1=0$. The other two are obtained from the condition ${\lambda }_2=c{\lambda }_1^3$, with $c\in {\mathbb{F}}_p^{\ast }$, and are defined by two simple equations involving the norm ${\lambda }_1^{p+1}$. The proof is based on the AGW criterion and on the study of a quartic curve naturally associated with the rational function induced on the unit circle ${\mu }_{p+1}$.