Rational points on certain families of symmetric equations

Wade Hindes

paper

Rational points on certain families of symmetric equations

arXiv:1403.0645

Abstract

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2 m} + a x^{m} + a y^{m} + y^{2 m} = b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe $X_{d} (Q)$ for certain $d \geq 3$ , where $X_{d} : T_{d} (x) + T_{d} (y) = 1$ and $T_{d}$ is the monic Chebychev polynomial of degree $d$ . Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus $3$ curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of $x^{4} - 4 x^{2} - 4 y^{2} + y^{4} = - 6$ .