Diophantine problems related to cyclic cubic and quartic fields

TL;DR

This paper investigates specific polynomial congruences connected to cyclic cubic and quartic fields, and computes all integer points on related genus 1 and 3 curves, advancing understanding of Diophantine problems in algebraic number theory.

Contribution

It introduces new polynomial congruences linked to cyclic cubic and quartic fields and provides comprehensive computations of integer points on related algebraic curves.

Findings

Solutions to the polynomial congruences are characterized.

All integer points on certain genus 1 and 3 curves are explicitly computed.

Results contribute to the understanding of Diophantine equations in algebraic number theory.

Abstract

We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.