Heights and Effective Theory of Quadratic Forms over Global Fields

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Mathematics (CMC)

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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 "size" (explicitly bounded), where the size is measured by a naive height function: the maximum of absolute values of the coordinates of the point in question; 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 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, have been obtained by Schlickewei over Q (1985) and by Vaaler over number fields (1987). More recently, there has also been 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 even over quaternion algebras. In this talk, I will give a survey of this lively area, starting from Cassels' original result and up until the recent developments.

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© Lenny Fukshansky

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