By Jean Pierre Serre

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

Example text

F o r the proof s e e B ourbaki, Alg . , ch. 8, §1 2 , no I, prop. 3. DEFINITION - For each prim e 1 let P1 be a rational l - adic rep re ­ s entation of K. The sy s te m ( Pl ) i s said t o be c ompatible if P l , Pl ' are compatible for any two prime s 1 , 1 ' . The sys tem ( p ) i s said 1 to be strictly c ompatible if the r e exists a finite subset S of L K such that: = {vi Pv = d. Th e n , for every viaS u Sl' Pl is (T) ha s rational coeffic ients . unramifie d at v and P v, Pl (b) P ( T ) = Pv l (T ) g v , S u Sl u 51' .

O th e induc ed homomorphi sm o f H(O ) into l (01) = Aut(V )· If p is an 1-adic representation of Gal(K/K) 1 o into H(Q ) ' one g et s by c ompo sition a linea r 1 - adic r ep r e s entation L q, c p : Gal(K /K) � Aut( V 1 ) . U sing the fa ct that the coeffi cient s of s I. th e c h a r acte r isti c p olynomial are c e nt ral func tion s , one s e e s that GL V AB E LIAN L -ADIC REPRESENT A TIONS 1-16 4>L is rational if p rational (K a number field). Of course, compatible representations in H give compatible linear representa­ tions.

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 " . ) .... ] ( ii ) I s it true that IITl( p ) and IITl( p ' ) are Lie g r oup s of the 1-13 J. - ADIC REPRESENT ATIONS More optimistically, is it true that there exists a same dimension? Li e alg eb r a Li e (I m (p l » 5.

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