Partial Descent on Hyperelliptic Curves and the Generalized Fermat Equation x^3+y^4+z^5=0

Abstract

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this factorization with the property that if it is empty then the curve C has no rational points. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with signature (3,4,5), which is unassailable via the previously existing methods.

References in corpus (2)

• Two-cover descent on hyperelliptic curves
• Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p

Cited by in corpus (6)

• The Generalised Fermat Equation x^2 + y^3 = z^15
• Descent on superelliptic curves
• Number of solutions to a special type of unit equations in two variables
• Generalised Fermat equation: a survey of solved cases
• Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality
• Baker's Explicit abc-Conjecture and applications