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

By Jean-Pierre Serre

This vintage publication comprises an creation to platforms of l-adic representations, an issue of significant significance in quantity concept and algebraic geometry, as mirrored through the astonishing 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 known as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out advanced multiplication, the most results of that is that a twin of the Galois crew (in the corresponding l-adic illustration) is "large."

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Xr ]–submodule of A[x1 , . . , xr ]s generated by ∪∞ i=1 Jn . Since A[x1 , . . , xr ] is noetherian we can find a finite number og generators (p1,1 , . . , p1,s ), . . , (pt,1 , . . , pt,s ) for P . We can choose the pi such that the pi,j have the same degree dj for i = 1, . . , s. Let m be the maximum of d1 , . . , dt . s Given an element l ∈ I n M ∩ N . We can write l = i=1 fi (a1 , . . , ar )mi , with (f1 , . . , fs ) ∈ Jn . consequently we get t (f1 , . . , fs ) = gj (x1 , . .

Then x is regular for M/I i M for i = 1, 2, . . i Moreover, if x is regular for M/I i M for i = 1, 2, . . and ∩∞ i=1 I M = 0, we have that x is regular for M . Proof. We show the first assertion by induction on i. For i = 1 the assertion holds by assumption. Assume that x is regular for M/I i M and that there is an m ∈ M such that xm ∈ I i+1 M . 11 Regular sequences 2 and since x is regular for I i M/I i+1 M we obtain that m ∈ I i+1 M , as we wanted to show. To prove the second assertion we take an m ∈ M .

Ar )mi , with (f1 , . . , fs ) ∈ Jn . consequently we get t (f1 , . . , fs ) = gj (x1 , . . , xr )(pj,1 , . . , pj,s ) j=1 with gj ∈ A[x1 , . . , xr ]. On the left hand side we have homogeneious polynomials of degree n. Consequently, we may, after possibly removing terms on the right hand side, assume that deg gj + dj = n for j = 1, . . , t and i = 1, . . , s. Then we have that s l= fi (a1 , . . , ar )mi = i=1 s t s gj (a1 , . . , ar ) j=1 pi,j (a1 , . . , ar )mi i=1 where i=1 pi,j (a1 , .

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