The field of definition of function fields and a problem in the reducibility of polynomials in two variables
Introduction Let K be a number field, O the ring of integers of K. Suppose we are given a projective curve Z, and a morphism :Z pr(C) (where pr(C) is the projective line) such that and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z)^/K(Pr(C))) (where K(Z) is the normal closure of K(Z) over K(Pr(C) ). In Section 1 we investigate (for our needs) the combinatorial and group theoretical aspects of the situation where
Two Problems Concerning Polynomials.
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How to do a p-descent on an elliptic curve
In this paper, we describe an algorithm that reduces the computation of the (full) p-Selmer group of an elliptic curve E over a number field to standard number field computations such as determining the (p-torsion of) the S-class group and a basis of the S-units modulo pth powers for a suitable set S of primes. In particular, we give a result reducing this set S of 'bad primes' to a very small set, which in many cases only contains the primes above p. As of today, this provides a feasible algorithm for performing a full 3-descent on an elliptic curve over Q, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of E[p] is favorable, simplifications are possible and p-descents for larger p are accessible even today. To demonstrate how the method works, several worked examples are included.