By Jean Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

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**Additional info for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)**

**Example text**

N S E G, X X E X (H» . = ---- n s (X ) for all (b ) n > 0 for e v e r y X E X (H) . X - The inj e c tivity of the map cP � e i s well -known (and doe s cP not depend on the c ommutativity of H) . T o p r ove s urj e c tivity , c on (i) s ider fir s t the c a s e whe r e e has the form e = � X e1 ) whe r e X Pr oof. is a full s e t of diffe r ent c onj ugat e s of a c har acte r X i s the s ubgr oup of G fixing X e = (* ) The fixe d field k X X � If G (X ) then s E G/ G (X ) o f G (X ) in X (H) .

X ) . l - ADIC R E PR E S E NT A T IONS i r r e ducible cha ra c te r s X of G generate a den s e sub spac e of Henc e the p r opo sition follows f r ozn leznzna 1. C (X) . - Let I-' be the Haa r zn ea sur e of G � I-' ( G) = l . Then a s e quenc e ( xn ) o f eleznent s o f X is I-' - e quidi stributed if and only if for any i r r e duc ible c ha racte r X of G , X f:. : X (x. ) n�CX) n i= l 1 = 0 This follow s frozn P rop . 2 and the following fact s : I-' ( X ) = 0 if X is i r r e duc ible f:.

Henc e , one of H the ac tion of � �A is th e g r oup alg ebra of G - h o m om o r ph i s m if A = A by linear ity , a homomorphism a : k[ X {H) ] whe r e k[X {H) ] let k be the k X {H) c an be identifie d with an H , and G X {H) ove r k . This is a is define d by s (�a X ) = �s (a ) s (X ) for a E k and X E X (H) . It is well - known X X X (line ar independenc e of cha r acte r s ) that a is inj e ctive . It is b ij e c tive if and only if H is a g r oup of multiplic ative type (cf. 1. 3 , re mark 2 ) .

### Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre

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