By Jean Pierre Serre
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Extra resources for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
Co ) . Con sequently, by the Abel summation trick ( d . [13 ] , p . 124 , p rop . ;. ;. co ) . ;. c X c o nc lude the as p ro of . ;. co , q. e. d. CHAPTER II TH E GROUPS � Throughout this chapte r , K denote s an algebraic number field. We a s s oc iate to K a pr oj e c t ive family (S ) of c ommutative alge m braic gr oups over Q , and we show that each Sm g ive s rise to a s tr ictly compatible system of r ational 1 - adic repre s entations of K . In the next chapter , we shall s e e that all " locally algebraic " ab elian rational repr e s entations are of the form de s cribed here .
Applying the o r e m 1 and propo s ition 2 , we obtain THEOREM 2 - T he element s xv (v &: L ) are equidistributed in X with r e spect to a mea sure I-' such that for any irreduc ible c harac te r X of G we have I-' ( X ) = c X COROLLAR Y - The elements xv (v &: L) a r e equidis tribute d for the normalize d Haa r mea sur e of G if and only if c = 0 fo r every X i r r e duc ible c haracte r X -F. 1 of G, i . e . , if and only if the L - functions relativ e to the non trivial i r r e du c ible character s of G are holomo rphic and non z e r o at s = 1 .
G U ) onto a den s e s ub g r oup of the d e c omp o s ition v ab . . ). ln G g r oup of v (r e s p . onto the lne r h a g r oup of v ,, and th at a unifo r m i z ing e l em ent f of K i s mapp e d onto the v v F r ob e n iu s c l a s s of v . v '. -' 1 If v S upp ( m ) and 1 , 1= 1 (a) = 1 , henc e p cr v th i s p r ov e s b ) . �) , a E Uv £ 1 (a ) = F or s uc h a v , the n E 1 an d we have and �) follows fr om �) . CORO LLARY c ompatible 1 - The r e pr e s entations E1 = 1; (a) £ if m o r e ov e r i s unr am i f i e d a t v ; 1 £ (f ) = E (f ) = l v v form a F v ; hence s y s tem of s t r i c tly - adic r e pr e s entation s with value s i n Sm W e als o s e e that the exc e ptional s e t of th i s s y stem i s c ontaine d in S upp ( m ) ; for an example whe r e it is diffe r ent fr om Supp ( m ) , s e e Exe r c i s e 2 .
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre