Classgroups and Hermitian Modules

1. Front Matter

   Pages I-XVII

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2. Preliminaries

   • A. Fröhlich

   Pages 1-19

3. Involution Algebras and the Hermitian Classgroup

   • A. Fröhlich

   Pages 20-77

4. Indecomposable Involution Algebras

   • A. Fröhlich

   Pages 78-116

5. Change of Order

   • A. Fröhlich

   Pages 117-145

6. Groups

   • A. Fröhlich

   Pages 146-197

7. Applications in Arithmetic

   • A. Fröhlich

   Pages 198-220

8. Back Matter

   Pages 221-226

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These notes are an expanded and updated version of a course of lectures which I gave at King's College London during the summer term 1979. The main topic is the Hermitian classgroup of orders, and in particular of group rings. Most of this work is published here for the first time. The primary motivation came from the connection with the Galois module structure of rings of algebraic integers. The principal aim was to lay the theoretical basis for attacking what may be called the "converse problem" of Galois module structure theory: to express the symplectic local and global root numbers and conductors as algebraic invariants. A previous edition of these notes was circulated privately among a few collaborators. Based on this, and following a partial solution of the problem by the author, Ph. Cassou-Nogues and M. Taylor succeeded in obtaining a complete solution. In a different direction J. Ritter published a paper, answering certain character theoretic questions raised in the earlier version. I myself disapprove of "secret circulation", but the pressure of other work led to a delay in publication; I hope this volume will make amends. One advantage of the delay is that the relevant recent work can be included. In a sense this is a companion volume to my recent Springer-Ergebnisse-Bericht, where the Hermitian theory was not dealt with. Our approach is via "Hom-groups", analogous to that followed in recent work on locally free classgroups.

• Invariant
• Volume
• algebra
• algebraic invariant
• arithmetic
• automorphism
• character
• cls
• group
• time
• matrix theory

• Mathematics Department, Imperial College, London, England

  A. Fröhlich

• Mathematics Department, Robinson College, Cambridge, England

  A. Fröhlich

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