Group Theory

Jean Pierre Serre's Abelian L-Adic Representations and Elliptic Curves (Advanced PDF

By Jean Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

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

Example text

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 .

An easy a rgument (cf. ch. III, 2 . 2 ) shows that p is almos t eve rywhe re unramifie d ( i . e . , if U v denot e s the g r oup of units at v , then p(U v ) = 1 fo r almost all v ) . Choo s e 1f v € K with v(1f v ) = 1. If p is unramified at v , then p ( 1f v ) depends only on v , and we s e t xv = A 1fv ) . W e make the following a s s umption: O ( * ) The homomo rphism p map s the g roup C of id�les of volume 1 onto G. (Recall that the volume of an id�le a = (av ) is defined as the produc t of the normaliz e d absolute values of its c omponent s a v , cf.

If h. = IT [a] a denote s a aEr c har acte r of T , then X E i s the s ubgr oup of tho s e n IT a (x) a = 1 , for all x E E . h. E X for which Exer c i s e s o that dim T = 2 . Let E b e the g r oup of units of K . Show that T is of diInens ion 2 (r e s p . 1) E if K i s iInag inary (re s p . r e al) . a . L e t K b e quadr atic ove r 0 , b . T ake for K a c ub ic field with one r e al place and one c om ­ plex one , and l e t again E b e its g roup of units (of r ank 1) . Show that dim T = 3 and diIn T E = 1.

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Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre


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