Points below a parabola in affine planes of prime order

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the set of points below the diagonal line. It is already surprising that a set in AG(2, p) defined in such a way can exhibit notable properties. In this paper, we present another unexpected result in a similar vein. We investigate the set of points lying below a parabola. We prove that in some sense, this set of points looks the same from all but two directions, despite having only one non-trivial automorphism. In addition, we study the sizes of these sets, and their intersection numbers with respect to lines.