By Jean Pierre Serre
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Extra info for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
U s ing the formula E X E that E i s a n alg eb r aic s ubgr oup of T . Let T E = E X E , one s e e s be the quotient g r oup T I E ; then T E is al s o a torus ove r O . Its characte r g roup X = X (T ) is the s ub g r oup of X = X (T ) c ons isting of tho s e charac E E n te r s which take the value 1 on E . 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 .
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? [ n o : easy co u nter-exam ples]. 4 . Let p, p ' be rational i, i' -adic r ep re s entation s of K which are c OITlpatible and s eITli - s iITlpl e . ( i ) If p is abelian ( i .
1 to p ' and to U = Ke r ( p) . ) Gene ralize this to },, - adic rep r e s e ntations (with r e spect to a numbe r field E). 2) Let p ( re sp . pI) be a rational J. -adic ( r e s p . representation of K, patible. If s E. G = of degree n. (s) (resp. -ADIC REPRESENTATIONS i-th coefficient of the characteristic polynomial of p(s) (resp. of p'(s». Let P(Xo , ... , Xn ) be a polynomial with rational coefficients, and let Xp (resp. Xp) be the set of s e: G such that P(a o (s ) , . ,an (s» = 0 (resp.
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre