By Jean-Pierre Serre
This vintage publication comprises an advent to platforms of l-adic representations, a subject of serious value in quantity conception and algebraic geometry, as mirrored by way of the fantastic 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 identical to the Galois staff (in the corresponding l-adic illustration) is "large."
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Extra info for Abelian l-adic representations and elliptic curves
Let __A__ be the fraction which corresponds to the p - qz factor p - qz, and let the fraction corresponding to the other factor S be equal to ~, hence, so that, according to section 39, p S ~ A p - qz + p S M (p - qz)S' M - AS Since these two fractions are equal M - AS must be (p - qz)S· divisible by p - qz and the quotient of M - AS by p - qz is equal to the polynomial function P. Since p - qz is a divisor of M - AS, M - AS vanishes when we let z = l!.. Therefore, if we substitute the constant l!..
In the case treated here the total degree of y and z is either m or n. 58. If in the equation giving the relationship between y and z there are three different combined powers, such that, the largest total degree differs from the middie total degree by the same amount as the middle total degree differs from the smallest, then y and z can be expressed in terms of the new variable x by solving a quadratic equation. If we let y = xz and then divide by the least power of z, then z can be expressed in terms of x by means of the quadratic formula.
47. If y = ~, then a new variable x, by which both z and y can be 39 expressed without radicals, is found in the following manner. Since both z and yare to be functions of x, whose expressions are not to contain radicals, it is clear that this can be done if we let ~ = bx. First we have y = bx and y and expressed both y a + bz = b 2 x 2 , so that z z by functions of ~, z becomes bx 2 - 1, so that = bx 2 - 1. Thus we have x without radicals, and since y bx. m + 48. If y = (a bz) n, then a new variable x by means of which y can be expressed without radicals, is found as follows.
Abelian l-adic representations and elliptic curves by Jean-Pierre Serre