During this booklet, award-winning writer Goro Shimura treats new components and offers suitable expository fabric in a transparent and readable variety. subject matters contain Witt's theorem and the Hasse precept on quadratic varieties, algebraic idea of Clifford algebras, spin teams, and spin representations. He additionally contains a few uncomplicated effects no longer without problems came across in other places. the 2 precept topics are: (1) Quadratic Diophantine equations; (2) Euler items and Eisenstein sequence on orthogonal teams and Clifford teams. the place to begin of the 1st subject matter is the results of Gauss that the variety of primitive representations of an integer because the sum of 3 squares is largely the category variety of primitive binary quadratic types. provided are a generalization of this truth for arbitrary quadratic varieties over algebraic quantity fields and numerous purposes. For the second one subject, the writer proves the lifestyles of the meromorphic continuation of a Euler product linked to a Hecke eigenform on a Clifford or an orthogonal team. a similar is finished for an Eisenstein sequence on the sort of workforce. past familiarity with algebraic quantity idea, the booklet is generally self-contained. a number of common evidence are said with references for unique proofs. Goro Shimura gained the 1996 Steele Prize for Lifetime fulfillment for "his very important and broad paintings on arithmetical geometry and automorphic forms".
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