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

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Let v € L , let 1 F. pv and let Fv be the Frobenius conjugacy cla s s of v in Aut( T1 (E» . The eigenvalue s of Fv are algebraic numbe r s ; when embedded into C they give c onjugate c omplex numbe r s / 7rv ;v with 1 7r v I = Nvl 2 . We may write then , 7r = ( Nv ) l / 2 e V -i4J v with 0 < 7r - 4J v - < On the othe r hand, let G = SU( 2 ) be the Lie group of 2 X 2 unitary matric e s with dete rminant 1 . Any element of the spac e X of c onjugacy clas s e s of G contains a unique matrix of the form i4J 0 (e "IP," 0 � 4J � The ima g e in X of the Haar measure of G � 0 e -l� 2 s in 2 4J d4J.

The abelian variety ) ha s "bad r educ tion ", cf . [3 2]. Que stions 1. Let P P be a rational i-adic rep r e s entation. Is it true that v, p ha s rational c o efficients f o r all v such that at v? P i s unraITlified A sOITlewhat s iITlila r que stion i s : I s any c OITlpatibl e sy steITl strictly c OITlpatible? 2. Can any rational i-adic r ep re s entation be obtained (by ten sor p r oduc t s , dir ect SUITlS, etc . ) froITl one s c OITling froITl i-adic c ohoITlology ? 3 . Given a rationa l i-adic r ep r e s entati on p o f K, and a p riITle i' , doe s the r e exist a rational i' -adic r ep r e s e ntation p ' of K c OITlpatible with p?

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. T HE GRO UP II- 3 5 m (F or more example s , s e e 3 . 3 . ) 1 . 3 . Enlarg ing g r oup s Let k be a fie ld and A a c ommutative alg eb r aic g r oup ove r k. Le t (* ) o � Y � Y � Y � 0 2 l 3 an exact s e quenc e of (ab s tract) c ommutative group s , with Y 3 finite . Let b e a h o m om o r ph i s m of Y 1 into the g r oup of k - r ational points of A.

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