Arithmetic of Quadratic Forms

• The Quadratic Reciprocity Law.- Arithmetic in an Algebraic Number Field.- Various Basic Theorems.- Algebras Over a Field.- Quadratic Forms.- Deeper Arithmetic of Quadratic Forms.- Quadratic Diophantine Equations.

From the reviews: "This impressive monograph is devoted to some aspects of the arithmetic theory of quadratic forms to which the author has made important contributions in recent years. ... Anyone with an interest in the book's topic will find this volume well worth his or her time." (Ch. Baxa, Monatshefte fur Mathematik, Vol. 169 (1), January, 2013) "In this book Goro Shimura presents some very interesting material in a clear and readable style. ... There is plenty to learn from an introduction to a highly technical subject by one of its leaders. ... The book will serve well as a basis for a couple of postgraduate courses. ... the material is essentially self-contained the exposition is often very technical ... ." (Peter Shiu, The Mathematical Gazette, Vol. 96 (536), July, 2012) "Beginners and experts ... have much to gain from the author's insights and perspectives presented in this book on the arithmetic theory of quadratic forms. ... In order to make the treatment of these topics accessible to readers with a general background in abstract algebra, the author includes preliminary material on algebraic number theory and the theory of semisimple algebras. Throughout the book, the presentation is concise and elegant." (A. G. Earnest, Mathematical Reviews, Issue 2011 m) "Goro Shimura is one of the world's premier arithmeticians, with his name attached to a number of marvelous things. ... it presents a lot more than its title suggests and does so magnificently. ... Shimura's Arithmetic of Quadratic Forms is another very important monograph by this fine scholar and, to use a hackneyed but apt phrase, will richly repay the reader who invests his time in a careful study of its pages." (Michael Berg, The Mathematical Association of America, August, 2010)