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

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On C(X), b e a R a d o n m ea s u r e o n X d. Bourbak i, Int. , chap. ( i . e . a c o ntinuo u s l in e a r f o r m III, The §l). s eq ue nce (x ) n is to be iJ. - e quidistributed, or iJ. - unif o rm l y distributed, if iJ. 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 ) .

P(a'o (s), . ,a'n (5» = 0). l of G (use Exer. 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.

G PI' P2 a r e semi - s impl e and have the same trace (T r .. P I = T r c P 2 ) , the n they are i s omo rphic . 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.

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