In this monograph, the notion of “forms” associated to Fermat equations, P_{n}^{r} : X_{1}^{r}+X_{2}^{r}+ … +X_{n}^{r} = 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 H^{1}(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 a_{1}X_{1}^{r}+a_{2}X_{2}^{r}+ … +a_{n}X_{n}^{r} = 0 are K/k-forms of Fermat equations P_{n}^{r}. A natural question arises: *Are there any* K/k-*forms of* P_{n}^{r} *(so-called “twisted Fermat equations), which are not diagonal?* To address this question, the author considers *all* forms of P_{n}^{r}, 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 H^{1}(G,A), when A is non-abelian, is carried out. For instance, if K is the separable closure of k, then A(P_{n}^{r}) is given by the wreath product W of the group μ_{r} of r-th roots of unity and the symmetric group S_{n}, and the K/k-forms are then given by classes in the cohomology group H^{1}(G_{k},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 P_{n}^{r}, characterized by a cohomology class θ[Q], then the cohomology of the hypersurface, and hence its zeta-function are completely determined by H^{*}_{et}θ[Q]. The heart of this monograph is to compute the automorphism on the l-adic cohomology group of X_{n}^{r} induced by an element in A(P_{n}^{r}). 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