Classgroups and Hermitian Modules by Albrecht Fröhlich

By Albrecht Fröhlich

These notes are an multiplied and up-to-date model of a process lectures which I gave at King's university London through the summer time time period 1979. the most subject is the Hermitian classgroup of orders, and particularly of team earrings. so much of this paintings is released right here for the 1st time. the first motivation got here from the relationship with the Galois module constitution of earrings of algebraic integers. The primary objective was once to put the theoretical foundation for attacking what can be referred to as the "converse challenge" of Galois module constitution thought: to specific the symplectic neighborhood and worldwide root numbers and conductors as algebraic invariants. a prior version of those notes was once circulated privately between a couple of collaborators. in response to this, and following a partial answer of the matter through the writer, Ph. Cassou-Nogues and M. Taylor succeeded in acquiring a whole answer. In a distinct course J. Ritter released a paper, answering sure personality theoretic questions raised within the prior model. i personally disapprove of "secret circulation", however the strain of different paintings ended in a hold up in booklet; i am hoping this quantity will make amends. One good thing about the hold up is that the appropriate fresh paintings might be integrated. In a feeling it is a significant other quantity to my fresh Springer-Ergebnisse-Bericht, the place the Hermitian conception was once now not handled. Our strategy is through "Hom-groups", analogous to that in fresh paintings on in the community unfastened classgroups.

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15) r-T 1A where is extended to A @F F c is a symplectic representation of via the first tensor factor. , the order of is even for symplectic X, where deg(x) deg(x) is the degree of T. 16) X §5. T and as Discriminants and the Hermitian classgroup We shall now introduce the Hermitian classgroup, this to be the group in which discriminants take their values. Throughout For a first informal discussion suppose that a ring. If {xi} (X,h) is a Hermitian A-module and A= A . 5 the element Pf(h(x. ,F' * Fc ) (the Pfaffian of discriminant matrix) belongs to s * , uniquely determined modulo Det A .

1 Finally take IT a. 12) 1 ~ the field of real numbers and D II the 42 Hamiltonian quaternion. Then we may take the only, and we get a. 1 to have values + 1 Pf (diag(a. l D)) X 1 q number of negatives entries, or in other words Pf (diag(a. 13) H. We consider a second example, the hyperbolic Hermitian plane over an involution algebra A2 /2 , A x A with basis o h(x,x) x h(y,y), (A,-). = The underlying module is (1,0) and h(x,y) y = (0,1) and h is given by h(y,x) . 15) r-T 1A where is extended to A @F F c is a symplectic representation of via the first tensor factor.

One leaving F elementwise fixed, we call the pair A over (A,-) an involution algebra(over F), but by abuse of notation often denote it just by A given. 2) F, 21 the subscript denoting the transposition. This defines an in- t volutory automorphism X~ X of KA,F which maps and commutes with the action of volutory automorphism f~f The restriction of to action of C = GA,F into itself and thus also defines an in- rlF' of Ho~ (KA F' ) where f(X) = f(X) F ' cent(A) is again an involution and the . 1 Proposition: preserves The map action.

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