The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves

TL;DR

This paper investigates the intersection of two Fermat hypersurfaces in projective 3-space over finite fields, analyzing their quotient curves and arithmetic properties using computational algebraic geometry techniques.

Contribution

It introduces an explicit computational approach to study quotient curves arising from Fermat hypersurface intersections, including their Jacobian decompositions and arithmetic invariants.

Findings

The Jacobian of the genus 2 quotient is (4,4)-split.

Explicit quotient curves are computed using Gröbner basis algorithms.

The p-rank, zeta function, and rational points are analyzed for the reduced curves.

Abstract

We study the intersection of two particular Fermat hypersurfaces in $\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its quotients. Explicit computation of the quotients is done using a Gr\"obner basis algorithm. We also study the $p$-rank, zeta function, and number of rational points, of the modulo $p$ reduction of the curve. We show that the Jacobian of the genus 2 quotient is $(4,4)$-split.