The two main themes of the book are

(1) quadratic Diophantine equations;

(2) Euler products and Eisenstein series on orthogonal groups and Clifford groups.

Whereas the latest chapters of the book contain new results, a substantial portion of it is devoted to expository material related to these themes, such as Witt's theorem and the Hasse principle on quadratic forms, algebraic theory of Clifford algebras, spin groups, and spin representations. The starting point of the first main theme is the result of Gauss that the number of primitive representations of an integer as the sum of three squares is essentially the class number of primitive binary quadratic forms. A generalization of this fact for arbitrary quadratic forms over algebraic number fields, as well as various applications are presented. As for the second theme, the existence of the meromorphic continuation of an Euler product associated with a Hecke eigenform on a Clifford or an orthogonal group is proved. The same is done for an Eisenstein series on such a group.

The book is practically self-contained, except that familiarity with algebraic number theory is assumed and several standard facts are stated without detailed proof, but with precise references.

Readership

Graduate students and research mathematicians interested in number theory and algebraic groups.