Planar Polynomials arising from Linearized polynomials

In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+A{x}^q+Bx)\in {\mathbb{F}}_{q^3}[x]$, with $A,B \in $\mathbb{F}${q}$\mathrm{.}Inparticularwecompletelyclassifythepairs$(A,B)\in $\mathbb{F}${q}^2$suchthat$f_{A,B}(x)$ is planar using connections with algebraic curves over finite fields.