A celebrated theorem of Cassels (1955) asserts that an integral quadratic form, which is isotropic over Q, has a non-trivial integral zero of small height; the bound is in terms of the height of the coefficient vector of the quadratic form. In the later years, analogues of Cassels’ result have been proved over other global fields: over number fields by Raghavan (1975), over rational function fields by Prestel (1987), and over algebraic function fields by Pfister (1997). Further extensions of Cassels’ theorem to small-height isotropic subspaces of a quadratic space, using the contemporary theory of height functions as developed by W. M. Schmidt, have been obtained by Schlickewei (1985), Schlickewei & Schmidt (1987), and Vaaler (1987, 1989). More recently, I have done some work on effective (with respect to height) decompositions of bilinear spaces, as well as further generalization of this theory to the situations with additional algebraic conditions and over quaternion algebras. In this talk, I will discuss several results in this area, starting from Cassels’ original theorem and up until my own recent work. Some of the results discussed are joint work with W. K. Chan and G. Henshaw.
Heights and effective theory of quadratic forms over global fields (NOTE: THURS)
Heights and effective theory of quadratic forms over global fields (NOTE: THURS)