On effective Witt decomposition and Cartan-Dieudonne theorem

Details On effective Witt decomposition and Cartan-Dieudonne theorem On effective Witt decomposition and Cartan-Dieudonne theorem

2005-01-19

arxiv.org/abs/math/0501282v2

Canadian Journal of Mathematics, Vol.59 (2007), No.6, pp.1284-1300

Mathematics

Number Theory

16 pages, revised and corrected version, to appear in Canadian Journal of Mathematics

Scientific paper

Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of Cartan-Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of small heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$.

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