Rational points on certain families of symmetric equations

摘要: We generalize the work of Dem'janenko and Silverman for Fermat quartics, effectively determining rational points on curves x2m + axm aym y2m = b whenever ranks some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe Xd(ℚ) certain d ≥ 3, where Xd : Td(x) Td(y) 1 Td is monic Chebychev polynomial degree d. Moreover, show how this later problem relates to orbit intersection problems in dynamics. Finally, construct a new family genus 3 which break Hasse principle, assuming parity conjecture, by specifying our results quadratic twists x4 - 4x2 4y2 y4 -6.