Arithmetic and geometry of the curve \(y^ 3+1=x^ 4\).

*(English)*Zbl 0838.14018We consider both the arithmetic and geometry of the curve in the title. Let two points of a curve be equivalent if the image of their difference in the Jacobian of the curve has finite order. An equivalence class is called a torsion packet. The Weierstrass points form a torsion packet and they are exactly the \(\mathbb{Q} (\zeta_{12})\)-rational points on this curve. The latter result is obtained from the fact that the Mordell-Weil group of the Jacobian over the field \(\mathbb{Q} (\zeta_{12})\) is finite. Since the Mordell-Weil group over the rationals is also finite, we can describe all solutions of the equation in fields of degree 3 or less over the rationals. In addition, we find bases for the 2- and 3-torsion of the Jacobian and describe an isogeny from the Jacobian to the product of three CM elliptic curves. The finiteness of the Mordell-Weil group was shown using a 3-descent on the Jacobian that did not make use of this isogeny.

Reviewer: E.F.Schaefer (Santa Clara)

##### MSC:

14G05 | Rational points |

14H40 | Jacobians, Prym varieties |

11D25 | Cubic and quartic Diophantine equations |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |