Arithmetic of Quadratic Forms
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This book can be divided into two parts. The ?rst part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. The raison dâË etre of the book is in the second part, and so let us ?rst explain the contents of the second part. There are two principal topics: (A) Classi?cation of quadratic forms; (B) Quadratic Diophantine equations. Topic (A) can be further divided into two types of theories: (a1) Classi?cation over an algebraic number ?eld; (a2) Classi?cation over the ring of algebraic integers. To classify a quadratic form ? over an algebraic number ?eld F, almost all previous authors followed the methods of Helmut Hasse. Namely, one ?rst takes ? in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely it was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to ? at each prime spot, which is the same as Eichlerâs, but we de?ne it in a di?erent and more direct way, using Cli?ord algebras. In Sections 27 and 28 we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number ?eld, which is included in class ?eld theory. We prove it when the base ?eld is the rational number ?eld to make the book self-contained in that case.
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Goro Shimura was born in Hamamatsu, Japan on February 23, 1930. He received a bachelor's degree in 1952 and a doctorate in 1958 from the University of Tokyo. He taught at the University of Tokyo and Osaka University before becoming a visiting professor at Princeton University in 1962. He was a professor at Princeton from 1964 until his retirement in 1999. As a mathematician, his insights provided the foundation for the proof of Fermat's Last Theorem and led to tools widely used in modern cryptography. He wrote more than 100 papers and books including one about Imari porcelain. His memoir, The Map of My Life, was published in 2008. He received several awards including a Guggenheim Fellowship in 1979, the Cole Prize for number theory in 1976, the Asahi Prize in 1991, and the American Mathematical Society's Steele Prize for lifetime achievement in 1996. He died on May 3, 2019 at the age of 89.
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