The fermat equation over quadratic fields

Solutions of the Cubic Fermat Equation in Quadratic Fields

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x + y = z in Q( √ d), conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.

The Generalized Fermat Equation in Function Fields

We show that if r > n!(n! 2) the set of solutions x, E C(t) of a Fern-rat equation 1; u,x; = 0, a, E C(t), is the union of at most n!“! families with an explicitly given simple structure. In particular, the number of projective solutions, up to rth roots of unity, of such an equation is either at most n!“’ or infinite. The proof uses the function held version of the abc-conjecture due to Mason,...

Twisted Fermat curves over totally real fields

The generalized Fermat equation

This article will be devoted to generalisations of Fermat’s equation x + y = z. Very soon after the Wiles and Taylor proof of Fermat’s Last Theorem, it was wondered what would happen if the exponents in the three term equation would be chosen differently. Or if coefficients other than 1 would be chosen. We discuss the reduction of the resolution of such equations to the determination of rationa...

The Pell Equation in Quadratic Fields

where 7 is a given integer of a quadratic field F, and integral solutions £, 77 are sought in F. It has been shown that equation (1) has an infinite number of solutions if and only if 7 is not totally negative when F is a real field, and 7 is not the square of an integer of F when F is imaginary. We now obtain the following result : Let 7 be such that equation (1) has an infinite number of solu...