The Wallis Products for Fermat Curves

Brauer Points on Fermat Curves

The Hasse principle is said to hold for a class of varieties over a number field K if for any variety X in the class, the set of rational points X(K) is non-empty whenever the set of adelic points X(AK) is non-empty. Manin [Man] observed that the failure of the Hasse principle can often be explained in terms of the Brauer group of X, Br(X). The product rule implies that X(K) must be contained i...

Twisted Fermat curves over totally real fields

Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

Abstract. Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(Ck), 1 ≤ k ≤ l−2, where Ck are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(Ck) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (l,...

Partial Descent on Hyperelliptic Curves and the Generalized Fermat Equation

Let C : y = f(x) be a hyperelliptic curve defined over Q. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f = f1f2 . . . fr. We shall define a “Selmer set” corresponding to this factorization with the property that if it is empty then C(Q) = ∅. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with si...

On the Frobenius Endomorphisms of Fermat and Artin-schreier Curves

The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermât and Artin-Schreier curves which is an analogue of the proof of Stickleberger's theorem. 1. Correspondences. The connection between cubic Fermât curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums. That one can compute the number of points on a Fermât curve ...