By Jean Pierre Serre
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Additional resources for Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)
Let TP be the set of v e 1:K at which p is un ramified, and for which the coefficients ao, .. , an of the characteristic polynomial of Fv, p satisfy the equation P (ao, , an) = O. l(Xp). • . • Representations with values in a linear algebraic group Let H be a linear algebraic group defined over a field k. If k ' is a commutative k-algebra, let H(k') denote the group of points of H with values in k'. Let A denote the coordinate ring (or "affine ring") of H. An element f A is said to be central if f(xy) = f(yx) for any x, y E: H(k') and any commutative k-algebra k' .
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. P(a'o (s), . ,a'n (5» = 0). l of G (use Exer.
Let U be a subs et of X who se boundary ha s iJ. -mea sure ze ro, and, for all n, let n be the numbe r of m < n such that ·x e U . Then U lim (nU l n) iJ. ( U ) . n-»co Let U O be the inte r ior of U . W e have iJ. ( U o ) = iJ. ( U) . Let o E > O . B y the definition o f iJ. (U ) the r e i s a continuous function IjJ E C (X) , 0 � 1jJ � l, wi th c/J = 0 on X - U O and iJ. ( c/J ) � y ( U) - E . < n I n we have Sinc e iJ. n ( cf» U = m A B E LIAN I. l. I. I. l (U) . I. (X - U) . I. (U ) < lim inf n I n, which implies the propo U s ition .
Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics) by Jean Pierre Serre