By Jean Pierre Serre
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Extra info for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
Then -1 (a) 1r 1 (a 1 ) ) . * By the lemma , £ 1 is tr ivial on K an d , henc e , define s a map s ince Sm (Q 1 ) i s totally dis c onnecte d (it is an C � S (Q 1 ) m ; 1 - adic Lie gr oup ) , the latter homomorphism i s trivial on the c on c recalled that C / D ab may be identified with the Galois gr oup G of the maximal ab elian extens ion of K . So we end up with a homomorphism 1 Gab - adic r epre s entation of K with i. e . with an � Sm (Q 1 ) ' 1: values in Sm (d. Chap. I, 2 . 3 ) . This r e pr e s entation is rati on al in the s en s e of Chapte r I, 2 .
Of a E. -adic representation is then a continuous homomorphism Gal(K /K} s � A where V is a finite-dimensional vector Aut(V}, spa ce over the local field E}... , can be defined in a way similar to the 1-adic case. Exe rc i s e s 1) Let p and p I be two rational, s emi - s imple , c ompatible r ep re s entations . Show that, if Im( p ) is finite , the same is true for Im( p l ) and that Ke r ( p) = K e r ( p ' ) . (Apply exe r . 3 of 1 . 1 to p ' and to U = Ke r ( p) . ) Gene ralize this to },, - adic rep r e s e ntations (with r e spect to a numbe r field E).
3 . Given a rationa l i-adic r ep r e s entati on p o f K, and a p riITle i' , doe s the r e exist a rational i' -adic r ep r e s e ntation p ' of K c OITlpatible with p? [ n o : easy co u nter-exam ples]. 4 . Let p, p ' be rational i, i' -adic r ep re s entation s of K which are c OITlpatible and s eITli - s iITlpl e . ( i ) If p is abelian ( i . e . , if IITl( p ) i s abelian) , i s it true th_a t p ' is abelian? ( W e shall s ee in c hapte r III that thi s is true at lea st if p is "loc ally algebraic " .
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre