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By Jean Pierre Serre

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

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Equidi s tribute d if and only if for any irr educ ible characte r X of G we have 1 n lim r; X (x . I. (X ) . l - ADIC R E PR E S E NT A T IONS i r r e ducible cha ra c te r s X of G generate a den s e sub spac e of Henc e the p r opo sition follows f r ozn leznzna 1. C (X) . - Let I-' be the Haa r zn ea sur e of G � I-' ( G) = l . Then a s e quenc e ( xn ) o f eleznent s o f X is I-' - e quidi stributed if and only if for any i r r e duc ible c ha racte r X of G , X f:. : X (x. ) n�CX) n i= l 1 = 0 This follow s frozn P rop .

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 ) . 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 ) , one has p £ (a) = = sgn (a ) IT p 'Y 00 P v P (a ) P Show that c) P1 and (a) = 'Y ' -1 a1 F =p . P d ) S h ow th at Chap . I, 1. 2. p1 c o m) in c i d e s with the char acter X 1 of = { 2 } and t:n = 1 . Show that the 2 gr oups F. , Cm , Tm , Sm coinc ide with tho s e of Exe r c ise I , hence In that the exceptional set of the c or r e sponding s ys tem is empty. (2 ) L et 2. 4. K = Q , Supp ( Linear repr e s entations of Sm We r e c all fir s t s ome well known fac t s on r e pr e s entat ion s .

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