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By Jean-Pierre Serre

This vintage publication comprises an creation to structures of l-adic representations, a subject matter of serious value in quantity conception and algebraic geometry, as mirrored through the unbelievable 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 without advanced multiplication, the most results of that's that a twin of the Galois team (in the corresponding l-adic illustration) is "large."

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Fng = 0, for some elements h,···, fn E R, then the module is R/(h, ... , fn). Here is an exact sequence view: An element m of a module M corresponds to a homomorphism from R to M, sending 1 to m. Thus, giving a set of elements {m a }aEA E M corresponds to giving a homomorphism cp from a direct sum G := RA of copies of R, indexed by A, to M, sending the a;th basis element to mao If the ma generate M, then cp is a surjection. The relations on the ma are the same as elements of the kernel of the map G ----+ M.

We set 7f(x) = y. 10, 7f is actually a polynomial map. In addition, we claim that 7f factors through the set X/G. Indeed, since R is invariant under g, we have R n mx = g(R n mx) = R n g(mx). Since g(mx) = mg-lx, this says that 7f(g-lX) = 7f(x); that is, 7f : X -+ Y factors through a map X/G -+ Y. Under good circumstances, the map X/G -+ Y is surjective. If we are in situation where R is a summand of A(X), as in the cases Hilbert treated, then for any maximal ideal n of R we have nA(X) =I- A(X).

To this end note that if X c pr is an algebraic set defined by homogeneous polynomial equations Fi(xo, ... i(XI, ... , x r ) = Fi(l, Xl, ... , Xr ) = 0; thus X n U is naturally an algebraic set in Ar. Every polynomial ! (Xl, ... , Xr ) may be written in the form F(l, xl, . , x r ) for some homogeneous polynomial F(xo, ... , xr). , and let F be the result of multiplying each homogeneous component of ! by a power of Xo to bring up its degree to d. More formally, we may write It follows that F(l,XI, ...

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