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On Witt's theorem for unimodular quadratic forms. II.

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Integral lattices and hyperbolic reflection groups

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Integral equivalence of vectors over local modular lattices.

We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces φ : E → E′ to an isometry φ V : V → V′ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. We also determine sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs. †Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his 70th birthday.

Some extensions of Witt's theorem†

Representations of finite groups

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.

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

Suppose G is a finite group of exponent n and X an irreducible character of G. In this note we give sufficient conditions for the existence of a minimal degree splitting field L with Q(X) C L C Q(gn).

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Minimal splitting fields in cyclotomic extensions

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp . If p is odd, [△] ∈ S(K) if and only if where c is the tame ramification index of k/Qp , k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Allan Trojan

Papers

The Integral Extension of Isometries of Quadratic Forms Over Local Fields

Minimal splitting fields in cyclotomic extensions

The Schur subgroup of a $p$-adic field