Forms of Fermat Equations and Their Zeta Functions

In this volume, an abstract theory of 'forms' is developed, thus providing a conceptually satisfying framework for the classification of forms of Fermat equations. The classical results on diagonal forms are extended to the broader class of all forms of Fermat varieties.The main topic is the study of forms of the Fermat equation over an arbitrary field K. Using Galois descent, all such forms are classified; particularly, a complete and explicit classification of all cubic binary equations is given. If K is a finite field containing the d-th roots of unity, the Galois representation on l-adic cohomology (and so in particular the zeta function) of the hypersurface associated with an arbitrary form of the Fermat equation of degree d is computed.

The Zeta Function; Galois Descent; Nonabelian Cohomology; Weil Cohomology Theories and l-Adic Cohomology; Classification of Forms; Forms of the Fermat Equation I; Binary Cubic Equations; Forms of the Fermat Equation II; Representations of Semidirect Products; The I-Adic Cohomology of Fermat Varieties; The Zeta Function of Forms of Fermat Equations.