By Jean-Pierre Serre

This vintage booklet includes an creation to structures of l-adic representations, an issue of significant value in quantity idea and algebraic geometry, as mirrored by way of the superb fresh advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one reveals a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now known as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out complicated multiplication, the most results of that is that clone of the Galois crew (in the corresponding l-adic illustration) is "large."

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**Extra resources for Abelian l-adic representations and elliptic curves**

**Sample text**

In the Boolean arrangement, every element is modular, so we may take any maximal chain. 3. Thus a maximal chain of modular elements is given by V < {x = O} < {x = y = O} < {O}. Not all elements of the braid arrangement are modular, but V < {Xl = X2} < {Xl = X2 = X3} < ... < {Xl = X2 = ... = Xi} = T is a maximal chain of modular elements. 34 Let A be an arrangement and let L Mobius function /-LA = /-L : L x L --t 7L. as follows: L(A). 2 The Mobius Function JL(X,X) = 1 ~x

By the inductive assumption, we have r(Y) = i - 1. Therefore we have r(Y V Hi) = r(Y) + r(Hi) - r(Y 1\ Hi) (i-l)+l-r(V)=i. This shows that the partition (1ft, ... , 1ft) is independent. Next let X E L \ {V}. Let j be the largest integer such that V = X 1\ X j . ) r(X V Xj) (r(X) + r(Xj ) - r(X 1\ Xj)) 1. This implies that X 1\ X j+! is a hyperplane belonging to A. Thus Ax n 1fj+! = {X 1\ Xj+d is a singleton. 88 that if A has a nice partition 11" = (11"1, ... ) and bi = 11I"il, then 1I"(A, t) = • II (1 + b;t).

Q(8) = xyz(x + y + z)(x + y - z)(x - y + z)(x - Y - z). The reader should check that 71'(A, t) = (1 + t)(1 + 3t)(1 + 3t) = 71'(8, t). However, these arrangements are not L-equivalent. 8 shows that A has two lines which are contained in four hyperplanes. These appear in the picture as the two common points on the line at infinity of the two sets of three parallel lines. 8 also shows that 8 has no such lines. The factorization of these Poincare polynomials is remarkable. Next we prove some general results to explain their factorization.