By Jean Pierre Serre
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Extra resources for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
Ln G g r oup of v (r e s p . onto the lne r h a g r oup of v ,, and th at a unifo r m i z ing e l em ent f of K i s mapp e d onto the v v F r ob e n iu s c l a s s of v . 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 .
X G rn Mor e pr e c is ely, let l Q (d time s ) . s e t of emb e dding s of K into Q ; each a E r extends phis m K � 0 � 0 , hen c e defin e s a morphism [a] : 0 The c ollec tion of all [a] 1 s g ive s the is omo rphism T I Q � Gm l Q X X G l . Mo r e ov e r , the [a] 1 s m Q o f the char ac ter g roup X (T ) = HomQ (T I Q ' Gm l Q ) of • • • r b e the to a homomor T I Q � Gm l Q . form a bas i s T. Note that th e Gal o i s g roup Gal (Q I 0 ) acts in a natur al way on X (T) , viz. by p e rmuting the [ a ] 1 s .
Then p, p' are said to be compatible if the re exists a finite subset S of � p' K such that are un ramified outs ide of S -and P , ( T) -for ( T) = P v,p v ,p p and (In othe r wor d s , the c haract e ri s tic polynomial s of the Frobenius element s are the s ame for p and p ' , at least for almost all v ' s . ) If p : Gal(K/ K) � Au t ( V ) i s a rational 1-adic representation of K, then V ha s a c ompo s ition s e r ie s V = V :JV :J . . /V. 1+ 1 1 (0 < i< q - l ) siznpl e (i. e . i r r e ducibl e ) .
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre