By Jean-Pierre Serre

This vintage ebook comprises an creation to structures of l-adic representations, a subject of significant significance in quantity conception and algebraic geometry, as mirrored by way 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 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 staff (in the corresponding l-adic illustration) is "large."

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**Example text**

Then (i) If Pi,.. P a e L, Pa+1>.. Pn). Pn) - Q-Sd-2(Pa+l-Pn)- Proof, (i) If F is homogeneous of degree d, and the curve D: (F = 0) meets L in points Pi,.. 9), I must have L c D, so that by the lemma, F = HF'; now since Pa+1>-. Pn ^ L» obviously F e Sd-i(P a +lv. Pn)(ii) is exactly the same. D. 32 §2 I. 6) Proposition. Let k bean infinite field, and P},.. Pg e P\ distinct points; suppose that no 4 of P^,.. Pg) = 2. Proof. For brevity, let me say that a set of points are conconic if they all lie on a nondegenerate conic.

An) e k is thought of as 'evaluating the function f at P '. Define a correspondence { ideals J c A } -¥-> { subsets X c A \ } by j ,_> v(J)= { P € Ank I f(P) = 0 forall f e j } . Definition. A subset X c A \ is an algebraic set if X = V(I) for some I. 3, I isfinitelygenerated. If I = (f i,.. r}, so that an algebraic set is just a locus of points satisfying a finite number of polynomial equations. If I = (0 is a principal ideal, then I usually write V(f) for V(I); this is of course the same thing as V: (f = 0) in the notation of §§1-2.

To see what this means geometrically, set Y = 1, to get the equation in affine coordinates (x, z) around (0,1,0): z = x 3 + axz2 4-bz3. This curve is approximated to a high degree of accuracy by z = x 3 : x3+ the behaviour is described by saying that C has an inflexion point at (0, 1, 0). More generally, an inflexion point P on a curve C is defined by the condition that there is a line L c P 2 k such that F|L has a zero of multiplicity > 3 at P (see Ex. 9)). It is not hard to interpret this in terms of the derivatives and second derivatives of the defining equations: for example, if the defining equation is y = f(x), then the condition for an inflexion point is simply d2f/dx 2(P) = 0; this corresponds in the diagram to the curve passing through a transition from being 'concave downwards' to being 'concave upwards'.