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Arithmetic and geometry of the curve \(y^ 3+1=x^ 4\). (English) Zbl 0838.14018
We 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.

MSC:
14G05 Rational points
14H40 Jacobians, Prym varieties
11D25 Cubic and quartic Diophantine equations
14H55 Riemann surfaces; Weierstrass points; gap sequences
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