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Arithmétique des Algèbres de Quaternions by Marie-France Vignéras (auth.)

By Marie-France Vignéras (auth.)

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Additional info for Arithmétique des Algèbres de Quaternions

Example text

1 les P(X) seulement . D'apr~s f c'est-~-dire maximalement symbole {l,e,~,~e} On ~ =~ ou suivante, Toute et {F~Lnr,~} e d~montr6 H F maximal. Calcul e F~ la r e m a r q u e . Comme sur que si . 2 H F~ que F[X] nr . + C ½ ( ~ a ) n +... n , et [K" :K "2] = 4 . Si e E R" - R "2 , caraet~rise est un L :K(~) nr c a r r ~A d a n s K , et , et ear ~ =-i sinon. 37 Table du symbole de Hilbert : b 1 e ~e DEFINITION. premier Soient ~ p . Le p un nombre symbole de 1 -1 On voit imm6diatement @p 6gal est lement a,b le au symbole , si p~2 (Corollaire que symbole de .

Contenant une R-module complets des : v ~ fix6e permet End(V) l'endomorphisme (x,y) deux L=M 2 sur V : elK +e2K , on r6seau L dimension que avec la m a t r i c e . Si dans V , on R-endomorphismes notera de L . (i) Les ordres parcourt id6aux : (i) V existe R maximaux r6seaux de les Soient . On pose aE les normaux parcourent PREUVE dans M(2,K) , ou dans V/K de V/K End(L,M) a symbole vectoriel de M(2,K) Si de 1 i a) (~ Legendre Hilbert = { espace d'identifier On 1 M(2,K) base base le de P DE ~e 1 : (a,b) 2 ETUDE ~ 1 sinon.

On EXEMPLE. A ~ des bases . On ordres ( f l ~ a , f 2 fib) ne c h a n g e pas de si l ' o n deux rem- la d i s t a n c e des ordres M(2,R) (~nR R ~RnR) est est l'intersection de ~ l'ordre et . d'Eichler. Un ordre d'Eichler deux maximaux de distance ordres niveau R~ n ~gal de niveau n Un ordre d'Eichler deux de reseaux L = flR+f2R h 6 End(L) et le l e m m e V est de complets de que que n d'Eichler de justifient peut aussi . Les ~ R (~ R R ) = @ = End(L) N End(M) l'on . h 6 flR+L~n suivant nRR ~ - n R ) R la f o r m e V M = flR+f2~nR tels R~ n .

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