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

By Jean-Pierre Serre

This vintage booklet includes an advent to structures of l-adic representations, a subject of serious value in quantity conception and algebraic geometry, as mirrored by means of the dazzling 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's that a dead ringer for the Galois team (in the corresponding l-adic illustration) is "large."

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Assume that the following is true: for all i = 1, 2, . . , m and for any two distinct monomials xa y b and xc y d in Γ(fi ) with c < a and b < d, there exists xr y s ∈ Γ(fj ) for some j such that the point (r, s) lies on the left hand side of the line through (a, b) and (c, d). Then I is a reduction of I ∗ . Prior to proving this theorem, we discuss several supporting lemmas. 4. Let k[u1 , u2 , . . , un ] be a k-algebra and consider its k-subalgebra k[η1 u1 + η2 u2 + · · · + ηn un ] for nonzero η1 , .

Fm generate the ideal. 3 to including the line connecting (a, b) and (c, d). 4. Consider the ideal I = (x4 , x2 y + xy 2 , y 4 ). f. 1]). It can be checked directly t hat II ∗ (I ∗ )2 . Thus, I is not a reduction of I ∗ . On the other hand, we know that K1 = (x4 + y 4 , x2 y + xy 2 ) is a reduction of I, also through a direct computation: K1 I 2 = I 3 . We observe that x4 , y 4 are in Γ(x4 + y 4 ) and the term x2 y +xy 2 is a combination of two monomials corresponding to two points on the left hand side of the line through (4, 0) and (0, 4).

Even) indices j < k, there exists an even (resp. odd) index i such that j < i < k. Since (a1 , b1 ), (a2 , b2 ), . , (an , bn ) are vertices of a convex graph, (ai , bi ) is on the left of the line through (aj , bj ) and (ak , bk ). 3, K is a reduction of K ∗ . Since K ∗ ⊆ I and since their graphs have the same convex hull, K ∗ is a reduction of I. Thus, K is a reduction of I. Furthermore, as pointed out in Introduction, since I is non-principal, a reduction of I is minimal if it is generated by two elements.

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