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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.
List of contents
I Preliminaries.-
1 Locally free modules and locally freely presented torsion modules.-
2 Determinants and the Hom language for classgroups.-
3 Supplement at infinity.- II Involution algebras and the Hermitian classgroup.-
1 Involution algebras and duality.-
2 Hermitian modules.-
3 Pfaffians of matrices.-
4 Pfaffians of algebras.-
5 Discriminants and the Hermitian classgroup.-
6 Some homomorphisms.-
7 Pulling back discriminants.-
8 Unimodular modules.-
8 Products.- III Indecomposable involution algebras.-
1 Dictionary.-
2 The map P.-
3 Discriminants once more.-
4 Norms of automorphisms.-
5 Unimodular classes once more.- IV Change of order.-
1 Going up.-
2 Going down.- V Groups.-
1 Characters.-
2 Character action. Ordinary theory.-
3 Character action. Hermitian theory.-
4 Special formulae.-
5 Special properties of the group ring involution.-
6 Some Frobenius modules.-
7 Some subgroups of the adelic Hermitian classgroup.- VI Applications in arithmetic.-
1 Local theory.-
2 The global discriminant.- Literature.- List of Theorems.- Some further notation.
