Abelian l-adic representations and elliptic curves by Jean-Pierre Serre

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By Jean-Pierre Serre

This vintage publication includes an creation to structures of l-adic representations, an issue of significant significance in quantity idea and algebraic geometry, as mirrored via the remarkable contemporary 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 unearths a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now referred to as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out advanced multiplication, the most results of that is that just like the Galois team (in the corresponding l-adic illustration) is "large."

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Extra resources for Abelian l-adic representations and elliptic curves

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This section includes a discussion of the chromatic polynomial, a precursor of the characteristic and Poincare polynomials. Algebras In Chapter 3 let K be a commutative ring. We construct certain algebras over K associated with A. 1. 2. The algebra A(A) is the quotient of the exterior algebra E(A) based on A by a homogeneous ideal I(A), A(A) = E(A)j I(A). This algebra is constructed using only L(A). In the literature A(A) is sometimes called the Orlik-Solomon algebra. It will reappear in Chapter 5 with a topological significance.

O An edge in the Hasse diagram connects X with Y if X < Y. If A is defined by a polynomial Q(A), it is sometimes convenient to label elements of L(A) by the equations they satisfy. 3. 8 The lattice L(A) of the Boolean arrangement. Let Hi = ker(xi)' Let 1= {i 1, ... , ip} where 1 ::; i1 < ... < ip ::; £. Let HI = Hi! n··· n Hip. The lattice L(A) consists of the 2£ subspaces HI for all subsets I. 9 The lattice L(A) of the braid arrangement is isomorphic to the partition lattice. Proof. Let I = {l, ...

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

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