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Forms of Fermat Equations and Their Zeta Functions

Lars Brünjes
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Noriko Yui
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In this monograph, the notion of “forms” associated to Fermat equations, Pnr : X1r+X2r+ … +Xnr = 0, is developed. The main tool in this endeavor is the Galois descent. If K/k is a Galois extension with Galois group G, and if X is an object defined over k, then every object Y defined over k which becomes isomorphic over K (called a K/k-form of X) defines a class in the Galois cohomology group H1(G,A(X)) where A(X) denotes the automorphism group of X over K. The idea of Galois descent is to deduce properties of Y from properties of X by “twisting” with this class. For instance, let K be the separable closure of k; diagonal equations a1X1r+a2X2r+ … +anXnr = 0 are K/k-forms of Fermat equations Pnr. A natural question arises: Are there any K/k-forms of Pnr (so-called “twisted Fermat equations), which are not diagonal? To address this question, the author considers all forms of Pnr, classifies them, and then studies them by the method of Galois descent. When k is a finite field, he proposes to compute the zeta-function of such forms.

The detailed study of the Galois cohomology groups H1(G,A), when A is non-abelian, is carried out. For instance, if K is the separable closure of k, then A(Pnr) is given by the wreath product W of the group μr of r-th roots of unity and the symmetric group Sn, and the K/k-forms are then given by classes in the cohomology group H1(Gk,W). As an application, classification results of forms are obtained; especially, a complete explicit classification of binary cubic forms is achieved.

If Q is a special form of the Fermat equation Pnr, characterized by a cohomology class θ[Q], then the cohomology of the hypersurface, and hence its zeta-function are completely determined by H*etθ[Q]. The heart of this monograph is to compute the automorphism on the l-adic cohomology group of Xnr induced by an element in A(Pnr). Using this method, the zeta-function of a binary cubic form is calculated. Also this method allows one to compute the zeta-function of (not necessarily cubic) forms of Fermat equations, and calculations are carried out in some examples.

The monograph is written in very friendly manner with numerous examples. It is a good addition to the book of Gouvêa and Yui on Arithmetic of Diagonal Hypersurfaces over Finite Fields. It is highly recommended to anyone who is interested in computing zeta-functions of twisted Fermat equations.

 Noriko Yui is Professor of Mathematics at Queen's University in Kingston, Ontario

* 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