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By Jean Pierre Serre

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Of 2. 2). l is normalized, i. e. l(G) = 1. Let TP be the set of v e 1:K at which p is un ramified, and for which the coefficients ao, .. , an of the characteristic polynomial of Fv, p satisfy the equation P (ao, , an) = O. l(Xp). • . • Representations with values in a linear algebraic group Let H be a linear algebraic group defined over a field k. If k ' is a commutative k-algebra, let H(k') denote the group of points of H with values in k'. Let A denote the coordinate ring (or "affine ring") of H.

N � iJ. ;> i . e . if iJ. ;;. iJ. (f) a s n � co fo r any n f E: C ( X ) . Note that this implies that iJ. is positiv e and of total mas s said co , 1. N o t e al s o that iJ. ;> iJ. ( f ) n iJ. (f) = m e a n s that n 1 lim 1:: f(x . ) n-»co n i= 1 1 LEMMA 1 - Let (c/J ) be a family of continuous functions on X with the property that their linear c ombinati ons a r e dense in C ( X ) . Sup pas e that, for all the s equenc e (p n (c/J » n>l ha s a limit . Then the sequenc e (xn ) is equidi stributed with re spect to some measure iJ.

A continuous homomorphism p: Gal(Ksl K) --:;:. -adic representation of K with values in H. -adic Lie group. -A DIC R E P R E S E NT A T IONS m eans for p VE t o b e unra mifi e d a t a plac e fin es th e Fr obe nius e l em en t F E: F is rati o na l v,p We say, as b e for e, (a ) i s a finite s e t S th ere ou tside S, (b ) if v w, p that p ¢ H(Q) of � I. K � K ; if wlv , one de­ an d it s conjugacy c la s s if such that i s unramified i s rational ove r Q. Tw o r a ti onal r e p r e s en t a t ions p, p ' (f or p r ime s 1 , 1' ) a r e said to be compatible if there exists a finite subset S of � such that p S, the c onjugacy cla s s F p v,p K a re unramifie d out s ide S and such that fo r any c entral ele ment f E A and any v E � - S we have f( F ) = f(F v , p ,) .

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