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This paper is the third part of the series "Spherical higher order Fourieranalysis over finite fields", aiming to develop the higher order Fourieranalysis method along spheres over finite fields, and to solve the geometricRamsey conjecture in the finite field setting.  In this paper, we prove an inverse theorem over finite field for sphericalGowers norms, i.e. a local Gowers norm supported on a sphere. We show that ifthe $(s+1)$-th spherical Gowers norm of a 1-bounded function$f\colon\mathbb{F}_{p}^{d}\to \mathbb{C}$ is at least $\epsilon$ and if $d$ issufficiently large depending only on $s$, then $f$ correlates on the spherewith a $p$-periodic $s$-step nilsequence, where the bounds for the complexityand correlation depend only on $d$ and $\epsilon$. This result will be used inlater parts of the series to prove the geometric Ramsey conjecture in thefinite field setting.

Spherical higher order Fourier analysis over finite fields III: a spherical Gowers inverse theorem