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By Jean Pierre Serre

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Extra resources for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)

Example text

Its characte r g roup X = X (T ) is the s ub g r oup of X = X (T ) c ons isting of tho s e charac E E n te r s which take the value 1 on E . If h. = IT [a] a denote s a aEr c har acte r of T , then X E i s the s ubgr oup of tho s e n IT a (x) a = 1 , for all x E E . h. E X for which Exer c i s e s o that dim T = 2 . Let E b e the g r oup of units of K . Show that T is of diInens ion 2 (r e s p . 1) E if K i s iInag inary (re s p . r e al) . a . L e t K b e quadr atic ove r 0 , b . T ake for K a c ub ic field with one r e al place and one c om ­ plex one , and l e t again E b e its g roup of units (of r ank 1) .

Exer c is e a) Let k ' b e a c ommutative k - alg eb r a , with Spe c (k l and 1 ) . ) k' f. 0 , and c onne c ted (i . e . k ' c ontain s exactly two idempotents : 0 Show the existenc e of an e xa c t s equenc e : 11 -6 AB E LIAN L - AD IC RE P R E S E N T A T IO NS ) ---7 B (k ' ) ---7 Y 3 ---7 0 ---7 A (k ' b ) What happens when Spe c (k ' ) i s not c onne c te d ? § 2 . C O NS T R UC T IO N 2. 1. 0 AND S T OF m m IdMe s and idMe s - c la s s e s We define d in Chapter I , 2 . 1 the s et � of finite plac e s of the K co numb e r fie ld K .

V '. -' 1 If v S upp ( m ) and 1 , 1= 1 (a) = 1 , henc e p cr v th i s p r ov e s b ) . �) , a E Uv £ 1 (a ) = F or s uc h a v , the n E 1 an d we have and �) follows fr om �) . CORO LLARY c ompatible 1 - The r e pr e s entations E1 = 1; (a) £ if m o r e ov e r i s unr am i f i e d a t v ; 1 £ (f ) = E (f ) = l v v form a F v ; hence s y s tem of s t r i c tly - adic r e pr e s entation s with value s i n Sm W e als o s e e that the exc e ptional s e t of th i s s y stem i s c ontaine d in S upp ( m ) ; for an example whe r e it is diffe r ent fr om Supp ( m ) , s e e Exe r c i s e 2 .

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