No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
An Injectivity Result For Hermitian Forms Over Local Orders
Abstract
Let Λ be a ring endowed with an involution a → ã. We say that two units a and b of Λ fixed under the involution are congruent if there exists an element u ∈ Λx such that a = ubũ. We denote by H(Λ) the set of congruence classes. In this paper we consider the case where Λ is an order with involution in a semisimple algebra A over a local field and study the question of whether the natural map H(Λ) → H(A) induced by inclusion is injective. We give sufficient conditions on the order Λ for this map to be injective and give applications to hermitian forms over group rings.
... Theorem 2.10 (Fainsilber, Morales [8]). Let k be a field which is complete for a discrete valuation and let R be its valuation ring. ...
... Compared with [8] we add the assumption of τ -smoothness to eliminate the restriction on the residual characteristic. The proof is almost identical. ...
The reduced norm-one group G of a central simple algebra is an inner form of
the special linear group, and an involution on the algebra induces an
automorphism of G. We study the action of such automorphisms in the cohomology
of arithmetic subgroups of G. The main result is a precise formula for
Lefschetz numbers of automorphisms induced by involutions of symplectic type.
Our approach is based on a careful study of the smoothness properties of group
schemes associated with orders in central simple algebras. Along the way we
also derive an adelic reformulation of Harder's Gauss-Bonnet Theorem.
... It follows that M u /K is weakly ramified by Proposition 1.4, hence its square-root of the inverse different admits a self-dual normal basis generator, by [2, Theorem 1] and [3,Corollary 4.7]. ...
... Let y denote the reduction modulo pO K of y ∈ O K , then T r K/Qp (η) = T r k/Fp (η) = 0, since η generates a normal basis, hence p divides d−1 j=0 x j . Therefore, (3). The inclusion exp(pv −p Z) ⊆ N(M u /K) will thus be a consequence of the next result. ...
We use new over-convergent p-adic exponential power series, inspired by work
of Pulita, to build self-dual normal basis generators for the square root of
the inverse different of certain abelian weakly ramified extensions of an
unramified extension K of Qp. These extensions, whose set we denote by M, are
the degree p subextensions over K of Mp,2, the maximal abelian totally, wildly
and weakly ramified extension of K, whose norm group contains p. Our
construction follows Pickett's, who dealt with the same set M of extensions of
K, but does not depend on the choice of a basis of the residue field k of K.
Instead it furnishes a one-to-one correspondence, commuting with the action of
the Galois group of K/Qp, from the projective space of k onto M. We describe
very precisely the norm group of the extensions in M. When K is not equal to
Qp, their compositum Mp,2 yields an interesting example of non abelian weakly
ramified extension of Qp, with Galois group isomorphic to a wreath product.
Finally we show that, with a slight modification, our over-convergent
exponential power series endow certain differential modules with a Frobenius
structure, generalising a result of Pulita. Unfortunately, they then lose the
property we need to build self-dual normal basis generators, hence the
desirable link between Galois module structure and differential modules is not
yet obtained.
... The first question is now completely answered. With L/K as above, no such basis will exist if [L : K] is even ([2, Theorem 6.1a]) and if [L : K] is odd then such a basis exists if and only if L/K is at most weakly ramified ( [7,Corollary 4.8] and [6,Theorem 1]). ...
... We fix an algebraic closure of K, denotedK, and let L/K be a finite Galois extension insideK with abelian Galois group G. Let O K (respectively, O L ) be the valuation ring of K (respectively, L) and let l and k be the residue fields of L and K, respectively. Recall that we define A L/K to be the unique fractional O L -ideal such that [7,Theorem 4.5] and [6, Theorem 1], we know that a self-dual integral normal basis for A L/K exists if and only if L/K is at most weakly ramified (this result is valid even if G is not abelian). In this section we describe how to construct such bases whenever they exist, providing G is abelian and that the residue field of K is not of characteristic 2. ...
Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that TrF/E(g(x), h(x)) = δg, h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist.
Now let K be a finite extension of ℚp, let L/K be a finite abelian Galois extension of odd degree and let be the valuation ring of L. We define AL/K to be the unique fractional -ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for AL/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.
... This is equivalent to the existence of a self-dual normal basis generator for L, i.e., an x ∈ L such that L = KG.x and T L/K (g(x), h(x)) = δ g,h . If M ⊂ KG is a free O K G-lattice, and is self-dual with respect to the restriction of l to O K G, then Fainsilber and Morales have proved that if |G| is odd, then (M, l) ∼ = (O K G, l) (see [6], Corollary 4.7). The square-root of the inverse different, A L/K , is a Galois module that is self-dual with respect to the trace form. ...
Let K be a finite extension of \Q_p, let L/K be a finite abelian Galois extension of odd degree and let \bo_L be the valuation ring of L. We define to be the unique fractional \bo_L-ideal with square equal to the inverse different of L/K. For p an odd prime and L/\Q_p contained in certain cyclotomic extensions, Erez has described integral normal bases for A_{L/\Q_p} that are self-dual with respect to the trace form. Assuming K/\Q_p to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.
We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.
We reduce a study of polarized abelian varieties over finite fields to the classification problem of skew-Hermitian modules over (possibly non-maximal) local orders. The main result of this paper gives a complete classification of these skew-Hermitian modules in the case when the ground ring is a dyadic non-maximal local order. Comment: 20 pages
In this paper we begin by considering the equivariant genus of quite arbitrary hermitian forms over a group ring Ok[G], where Ok is the ring of integers in a number field K and G is an abelian group of odd order. The result we obtain is then applied to the case where G is the Galois group of a tamely ramified extension and the form is the one obtained by restricting the bilinear trace form to the ring OE of integers in E. More precisely let be the unique fractional ideal in E whose square is the inverse different of the extension ; then we construct a locally free ideal in OK[G] such that and we show that when equipped with the multiplication form tG on K[G], then () lies in the equivariant genus of (). Finally we show that when K = , then () (respectively ()) is actually isometric to () (respectively (Ok[G], tG)).
During the last few years several papers concerned with the foundations of the theory of quadratic forms over arbitrary rings with involution have appeared. It is not necessary to give detailed references, in particular one thinks of the well known work of Bak [l], Bass [3], Karoubi, Knebusch [ll, 121, Ranicki, Vaserstein, and C. T. C. Wall. During the same period a number of problems quite similar to those occuring in the theory of quadratic forms were discussed, which, however, did not fit in the formalism developed so far. For example, one thinks of problems like the classification of pairs of forms, of sesquilinear forms, isometries , quadratic spaces with systems of subspaces, and also of quadratic forms over schemes, see e.g. [12, 13, 20, 22, 231. This situation called for a more general foundation of the theory of quadratic and hermitian forms. In this paper we try to give this foundation. Our basic object is an additive category &! together wit a duality functor *: & + A. In this situation one can define the most important notions of the theory of quadratic forms. Under suitable finiteness conditions one can prove a Krull-Schmidt theorem which is a sharpening of the classical Witt theorem. This result is basic for applications to the problems mentioned above. A preliminary version of this material is contained in [ 171. As just one application we discuss the classification of quadratic spaces with four subspaces. We hope that this disucssion will show clearly how one can solve a number of important classification problems of linear algebra. More applications can be found in [15, 16, 17, 22, 241. In the second part of the paper we discuss hermitian (not quadratic) forms in an abelian category. In an abelian category one has more structure, in particular one can introduce the notion of orthogonality. This allows one to introduce Grothendieck and Witt groups analogous to the G-groups in linear algebraic Ktheory which are obtained by factoring out exact sequences. As a basic result a Jordan-Holder theorem is proved for categories where all objects are of finite length. Using this theorem the computation of the Grothendieck group is 264
The notion of module together with many other concepts in abstract algebra we owe to Dedekind [2]. He recognized that the ring of integers O K of a number field was a free Z -module. When the extension K/F is Galois, it is known that K has an algebraic normal basis over F . A fractional ideal of K is a Galois module if and only if it is an ambiguous ideal. Hilbert [4, §§105-112] used the existence of a normal basis for certain rings of integers to develop the theory of root numbers — their decomposition already having been studied by Kummer.
The book of involutions
- M.-A Knus
- A Merkurev
- M Rost
- J.-P Tignol
M.-A. Knus, A. Merkurev, M. Rost, and J.-P. Tignol, The book of involutions, to appear.