Explicit Kummer Theory for Elliptic Curves

Division in modules and Kummer theory

“…This result follows from Theorem 5.10, which is essentially an application of Theorem 5.3, which in turn is a generalization of [17,Theorem 5.9]. The results on Galois representations needed to apply this general theorem are mostly taken from [9], and it can be easily seen that the given bounds only depend on the ℓ-adic representations, so that the constant c of our main theorem is effectively computable.…”

“…In the case of elliptic curves, one may hope to obtain an explicit version of this result. Indeed the results of [9] and [17] provide such a statement under the assumption that End K (G) = Z, and they show that an effective bound depends only on the abelian group structure of A and on the ℓ-adic Galois representations associated with the torsion of G for every prime ℓ.…”

Some uniform bounds for elliptic curves over $\mathbb Q$

“…Our numerical estimate on the exponent of H 1 (G ∞ , E[N ]) is nowhere near as sharp as the corresponding bounds for the special case N = ℓ k , but notice that (unlike that case) it is not a priori clear that a uniform bound should even exist. We had in fact already shown the existence of such a bound in [38], but the result was not effective.…”

“…Finally, (4) was our original motivation for the work done in this paper: we had already shown a similar result in [38], but (lacking all the previous information (1), ( 2), (3)) we could not make it explicit, or in fact even effective. With all the preliminary work done in [38] and in the other sections of this paper, the desired result on Kummer extensions is now easy to prove.…”

“…We remark that we have chosen to formulate our bounds in terms of divisibility: we prove that multiplication by a suitable universal constant e kills the abelian group H 1 (G ∞ , E[N ]), and therefore the exponent of this group divides e. The numerical constant would be much smaller if we instead formulated the result as an inequality (that is, if we were content with knowing that the exponent of H 1 (G ∞ , E[N ]) does not exceed a certain constant e ′ ), but we feel that our version will be more useful in applications. In particular, we would like to stress that -even ignoring the non-effective parts of the argument -the ideas of [38] would lead to a (divisibility) bound for H 1 (G ∞ , E[N ]) involving primes up to several millions, while the value of e that we find with the new, more streamlined proof given in the present paper is only divisible by the primes up to 11 (which, as we show in Section 7, all need to appear as factors of e). In other words, while our constant e is probably not optimal, it is at least supported on the correct set of primes.…”