By Jean Pierre Serre
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Extra resources for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
2 . The c on s tr uc t i on of 1 . 3 now yields a O - algebraic gr oup Sm and a gr oup homomorphism with an alg ebr aic morphis m Tm � Sm £: I m � S (0) . m The s equenc e is exact (Cm be ing identifie d with the c or r e s ponding c ons tant alge br aic group) and the diag r am (** ) 1 � Tm (0 ) � Sm (0 ) � Cm � 1 is c ommutative . Remark Let m ' be another modulus ; a s s ume m ' :: m , i . e . > m if v E Supp ( m ) . From the in Supp ( m ' ) :J Supp ( m ) and mv' v clus ion Um , e Um one de duc e s maps T m , � Tm and � Im whence a morphism S m � Sm Henc e the S m ' s ' ' form a projective s ystem ; the ir limit is a p r oalgebr aic gr oup ove r 0, Im extens ion of the pr ofinite gr oup CI D Exe r c is e s =� Cm by a torus .
1 , s e e [13 J, c hap . VII . '" Remark This example ( e s s entially due to Hecke ) is given in Lang (loc . c it. , ch. VIII, §5 ) exc ept that Lang ha s replaced the c ondition ( * ) by the c ondition "p is surj e c tiv e " , which is insuffic ient. 1y distr ibute d m o dul o 1 ; how ev e r , 1 -2 5 l -ADIC REPRESENTATIONS one knows that thi s sequenc e i s not uniformly di stributed for any mea sure on R/ Z (d. P olya -Sze gtl [ 2 2 ] , p . 1 7 9 -18 0 ) . 3 . ( Conj e c tural example ) . Let E b e an elliptic curve defined ove r a numbe r field K and let L be the s e t of finite plac e s v of K suc h that E ha s g ood reduction at v, d .
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.
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre