Explicit \(4\)-descents on an elliptic curve.

*(English)*Zbl 0873.11036From the authors’ introduction: “The authors investigate how to find generators of an elliptic curve \(E (\mathbb{Q})\) modulo \(2E (\mathbb{Q})\) defined over \(\mathbb{Q}\). As is usual, they can reduce this to the study of a certain homogeneous space
\[
y^2= f(x,1), \tag{1}
\]
where \(f(X,Z)\) is a binary quartic form with integer coefficients. One wishes to know whether equation (1) has a \(\mathbb{Q}\)-rational point and if so to exhibit one. One can often show that equation (1) has no \(\mathbb{Q}\)-rational point by local methods. However, even if (1) is everywhere locally soluble, it does not follow necessarily that a \(\mathbb{Q}\)-rational point exists; this failure of the “Hasse principle” is well-known and gives rise to an element of the Tate-Shafarevich group.”

The difficulty in practice arises therefore when a homogeneous space is everywhere locally soluble, but we cannot decide whether or not a \(\mathbb{Q}\)-rational point exists on it; this may happen either because such a point does not actually exist but we have serious difficulties in proving this fact, or because it does exist, but has a very large height (which means that it is impossible to discover this point by brute force computations). To deal with such troublesome homogeneous spaces, the authors consider further descent on equation (1) and propose an explicit method suitable for machine calculation. We quote again from the Introduction:

“This has been done before in the literature for special types of elliptic curves. However, we could find no general account which was of use for systematic machine computations. We explain an explicit method for performing such further descent and we show this is equivalent to constructing elements of order dividing 4 in the Tate-Shafarevich group of the elliptic curve.”

After the whole method is developed (in which some sophistication is, probably, unavoidable), the paper ends up with a section, labeled “Examples”. This occupies less than one page and contains just one example with as few details as possible. In the reviewer’s opinion, this is a drawback of the paper.

The difficulty in practice arises therefore when a homogeneous space is everywhere locally soluble, but we cannot decide whether or not a \(\mathbb{Q}\)-rational point exists on it; this may happen either because such a point does not actually exist but we have serious difficulties in proving this fact, or because it does exist, but has a very large height (which means that it is impossible to discover this point by brute force computations). To deal with such troublesome homogeneous spaces, the authors consider further descent on equation (1) and propose an explicit method suitable for machine calculation. We quote again from the Introduction:

“This has been done before in the literature for special types of elliptic curves. However, we could find no general account which was of use for systematic machine computations. We explain an explicit method for performing such further descent and we show this is equivalent to constructing elements of order dividing 4 in the Tate-Shafarevich group of the elliptic curve.”

After the whole method is developed (in which some sophistication is, probably, unavoidable), the paper ends up with a section, labeled “Examples”. This occupies less than one page and contains just one example with as few details as possible. In the reviewer’s opinion, this is a drawback of the paper.

Reviewer: N.Tzanakis (Iraklion)