Martin J. Taylor

On Fröhlich's conjecture for rings of integers of tame extensions

where ] is the complex conjugate of Z. The constant W(Z) is called the Artin root number of ;/. In [ M l ] J. Martinet showed that OM is not always a free ZF-module when N = Q and when F is a quaternion group of order 8. Subsequently, in [F5], A. Frtihlich showed that the class of ~ u , in this special case, is determined by the sign of the Artin root number of the irreducible sympletic character of/2. We now return to the general case and let CI(TIF) be the class group of locally free ZF-modules. Given a locally free 2~F-module X, we denote its class by (X). Philippe Cassou-Nogu6s defined a class t(W) in CI (~F) which is defined solely in terms of the values of Artin root numbers of sympletic characters (this class will be defined in w After numerous computations by Fr6hlich et al. (cf. [F6], [C], [T3] for instance), Fr6hlich conjectured the result given below which we prove in this paper.

Group Rings and Class Groups

I Some general facts.- II Some notes on representation theory.- III The leading coefficient of units.- IV Class sum correspondence.- V More on the class sum correspondence.- VI Subgroup rigidity.- VII Global units.- VIII Locally isomorphic group rings.- IX Zassenhaus conjecture.- X Variations of the Zassenhaus conjecture.- XI Group Extensions.- XII Class sums of p-elements.- XIII Clifford theory revisited.- XIV Examples.- I Introduction and Review of the Tame Case.- II Hopf Orders.- III Principal Hornogeneous Spaces.- IV Arithmetic Applications:- The Cyclotomic Case.- V Arithmetic Applications:- The Elliptic Case.- References.

A logarithmic approach to classgroups of integral group rings

Let 1 be a fixed rational prime number. We denote the ring of rational l-adic integers by Z, , and the rational Z-adic field by Q, . For any ring R (always with a 1) we denote the group of units of R by Ii*. For a finite group r, FrGhlich has defined a group Det(Br~*) (cf. p. 383 of [ZJ). We give a precise definition of this group below, though essentially Det(&I’*) can be thought of as the reduced norms of units of the local group ring &r. This group plays a vital role in the description of various classgroups of modules over ZI’. The aim of this paper is to give a new and beautiful description of Det(&P) in terms of an “integral logarithm” obtained by use of Adams operations. In particular this description will enable us to answer certain questions concerning the Galois cohomology of such groups. This is very important because such cohomological questions play a crucial part in trying to describe the Galois module structure of rings of integers (cf. (13)). Before we can proceed to state the main theorems and give the new logarithmic description, we must introduce the concept of a Galois order over Zr , and then define Det(&r*). Let K/Q be a finite, unramified extension, let OK be the ring of integers of K and let A = Gal(K,Q). If A is a finite group with A ~4 A, then we define the Galois order over Zr

L-functions and arithmetic

1. Descent theory and finiteness results on the Tate-Safarevic group of elliptic curves Gross and Rubin 2. Automorphic L-functions Arthur and Gelbart 3. Beilinson conjectures Deninger and Scholl 4. Hilbert modular forms Ribet R. Taylor and Wiles 5. Iwasawa theory of motives Coates and Greenberg 6. p-adic cohomology Bloch and Fontaine 7. -adic representations attached to automorphic forms Clozel 8. L-functions and Galois modules Chinburg, Frohlich and M. Taylor.

Hermitian modules in Galois extensions of number fields and Adams operations

0. Introduction and statement of results 1. Some definitions and notation 2. The module M(N/K) 2.1. The group algebra FA 2.2. Determinants 2.3. Resolvends and resolvents 2.4. Adams operations 2.5. Valuation results; the integers v(8) 3. Semilocal isometries 3.1. Local tamely ramified Galois extensions 3.2. Local isometries 4. Double coset description of LI and minus determinants 4.1. Minus determinants, the invariant d 5. Strong approximation 6. Proof of the Main Theorem 7. The abelian case Appendix References

Principal Homogeneous Spaces

As before, let K be the field of fractions of a Dedekind domain \( mathfrak{D} \) of characteristic 0, and let G be a finite abelian group. We will continue to work with Hopf orders \( mathfrak{A} \) in A = KG and B = Map(G,K). In this chapter we will be concerned with the objects on which a Hopf order in A acts. Rather than studying all \( mathfrak{A} \) -modules, we will make use of the comultiplication in \( mathfrak{A} \) by considering only those \( mathfrak{A} \) -modules which have the structure of an \( mathfrak{D} \) -algebra, and are in fact “twisted” versions of \( mathfrak{B} = \mathfrak{A}D \) . These objects are the principal homogeneous spaces for the Hopf order \( mathfrak{B} \) , and the set of isomorphism classes of principal homogeneous spaces can be given the structure of an abelian group \( PH(\mathfrak{B}) \) . As in the previous chapter, we will first work at the level of K-algebras, and then see how the theory lifts to integral level. We shall then construct a group homomorphism ψ from \( PH(\mathfrak{B}) \) to the locally free classgroup \( C1(\mathfrak{A}) \) . Finally, in the case that G is cyclic of order p and K contains a primitive pth root of unity, we use Kummer theory to give an explicit description of \( PH(\mathfrak{B}) \) and of the kernel of ψ.

The Galois module structure of certain arithmetic principal homogeneous spaces

In this paper we obtain Galois module results for rings of integers of certain abelian extensions of cyclotomic extensions and of division fields for elliptic curves with complex multiplication. In the cyclotomic case we consider the rings of integers of non-ramified extensions, and in the elliptic case we are concerned with extensions arising from the Selmer group of the curve. In each case we can describe those rings of integers which are free Galois modules, by means of L-function congruences. This then is a further instance of how Galois module structure is dominated by L-functions, with the classic case being the tame theory (see [F], [T3]), where it is the root numbers of the sympletic characters of the Galois group which determine the Galois module structure. In broad terms, the results of this paper arise from marrying the algebraic results in [Tl] and certain powerful results from Iwasawa theory (both cyclotomic and elliptic). For aesthetic reasons, as well as for the sake of brevity, we introduce notation which will simultaneously cover both the cyclotomic and the elliptic case. We begin by defining the notation for

Hasse-Witt invariants of symmetric complexes: an example from geometry

Abstract Jardine has defined Hasse–Witt invariants for symmetric bundles over schemes. This definition can be extended to symmetric complexes, that is symmetric objects in the derived category of bounded complexes of vector bundles over a scheme. In this Note we show how one can use these generalized invariants to give a neater proof of a comparison result on Hasse–Witt invariants of symmetric bundles attached to tame coverings of schemes. To cite this article: P. Cassou-Nogues et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 839–842.

The classifying topos of a group scheme and invariants of symmetric bundles

Let

Duality and Hermitian Galois Module Structure

Y