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The Integral Extension of Isometries of Quadratic Forms Over Local Fields

Published online by Cambridge University Press:  20 November 2018

Allan Trojan
Allan Trojan*
Affiliation:
Massachusetts Institute of Technology and McGill University
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Let F be a local field with ring of integers 0 and prime ideal π0. If V is a vector space over F, a lattice L in F is defined as an 0-module in the vector space V with the property that the elements of L have bounded denominators in the basis for V. If V is, in addition, a quadratic space, the lattice L then has a quadratic structure superimposed on it. Two lattices on V are then said to be isometric if there is an isometry of V that maps one onto the other.

In this paper, we consider the following problem: given two elements, v and w, of the lattice L over the regular quadratic space V, find necessary and sufficient conditions for the existence of an isometry on L that maps v onto w.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 920 - 942
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. O'Meara, O. T., Introduction to quadratic forms (New York, 1963).Google Scholar
2. O'Meara, O. T., The integral representations of quadratic forms over local fields, Amer. J. Math., 80 (1958), 843–878.Google Scholar
3. Rosenzweig, S., An analogy of Witt's theorem for modules over the ring of p-adic integers, Doctoral thesis, Mass. Inst. Techn., 1958 (unpublished).Google Scholar