Integral quadratic forms / G. L. Watson

Bibliogr. p. 141-142. Index

This monograph is chiefly concerned with equivalence and representation of integers by quadratic forms with integral coefficients. A background of only elementary number theory and the rudiments of matrix theory is assumed. The methods are arithmetical and at times most ingenious. The first five chapters are mostly classical. They are concerned with reduction, the rational invariants (Hilbert and Hasse symbols), p-adic equivalence from the point of view of congruences, the congruence class, and genus. The fifth chapter contains the author's beautiful result that an indefinite form in four or more variables represents properly an integer a if it represents it properly modulo d.
Chapter 6, which deals with rational transformations and automorphs, is largely preparation for the following chapter. In chapter 7, the spinor norm of Eichler is defined in arithmetical terms. ... Chapter 8 is concerned with factorization of the general rational automorph into reflexions with denominators prime to a given odd integer m. In particular, every rational automorph of a form with discriminant d, whose weight is prime to d, has the same norm as some product of rational reflexions of f, each reflexion having denominator prime to d.
Though in a number of spots this tract is very difficult reading, it is a notable contribution to the literature of quadratic forms and brings one to the threshold of much that is still unknown in the subject. (MathSciNet)