International Congress of Mathematicians

192 B. Mazur K. Rubin6. Orthogonal A-modulesIn this final section we introduce a purely algebraic template which, when it"fits", gives rise to many of the properties conjectured in the previous sections.Keep the notation of the previous sections. In particular r : A —t A is theinvolution of A induced by complex conjugation on K, and if V is a A-module,then V^ denotes V with A-module structure obtained by composition with r. LetV* = HOIIIA(V, A). If V is a free A-module of rank r, then detA(V' T ) will denotethe r-th exterior power of V and a r-gauge on V is a A-isomorphism between thefree A-modules of rank onet v : det A (F*) ^det A (V'( T) )or equivalently an isomorphism detA(V) ® detA(V r(T^) -^ A.By an orthogonal A-module we mean a free A-module V with semi-linearr-structure endowed with a r-gauge tv and a A-bilinear r-Hermitian pairing (Definition2)IT : V ®A V (T) —y A.Viewing n as a A-linear map V^ —^ V*, the compositiont v o det A (7r) : det A (V'( T) ) —y det A (F*) —y det A (V'( T) )must be multiplication by an element disc(V) £ A that we call the discriminantof the orthogonal A-module V. We further assume that disc(V) ^ 0, and we defineM = M(V, ix) to be the cokernel of the (injective) map ix : V^ —^ V*, so we have0 —• V {T) —• V* —• M —• 0. (6.1)If K C F c KQO, recall that IF = ker{A -» Ap} and defineV(F) := {x £ V : TT(X,V {T) ) C I F }/IFF = ker{F ® A A F ^ (V {T) )* ® A A F }and similarly V^ (F) := ker{V r M ®\ F _+ V*®A F }. Any lift f of r to Ga^K^/Q)induces an isomorphism V(F) —^ V^(F). From (6.1) we obtain0 —• V {T) (F) —• V {T) ® A A F —• V* ®A Ap —• M ®A Ap —• 0 (6.2)and (applying Hom( • , Ap) and using the Hermitian property of n)We have an induced pairingV(F) ~ Hom AF (M ®A Ap, A F ). (6.3)np : V^(F) ® AF V(F) —• I F /F F ,which we call the F-derived pairing. If F is stable under complex conjugation then\/( T i(F) is canonically isomorphic to V(F)^T' > and -Kp is r-Hermitian.