On the average behaviour of the rank of the curves \(y^ 2= x^ 3 +k\). (Sur le comportement en moyenne du rang des courbes \(y^ 2= x^ 3 +k\).) (French) Zbl 0814.11034

Fouvry, E.

David, Sinnou (ed.), Séminaire de théorie des nombres, Paris, France, 1990-1991. Basel: Birkhäuser. Prog. Math. 108, 61-84 (1993).

Let \(r(k)\) be the rank (over \(\mathbb{Q}\)) of the elliptic curve \(y^ 2 = x^ 3 + k\). It is shown that the average of \(3^{r(k)/2}\) is bounded. It follows that the proportion of such curves with rank at least \(R\) is \(O(3^{-R/2})\). The proof in fact shows that the corresponding results hold when \(r(k)\) is replaced by \(s(k)\), the Selmer rank, as calculated via 3-descents. Most of the work in the proof goes into estimating \(s(k)\) in terms of the 3-rank of the idele class group of \(\mathbb{Q}(\sqrt{k})\). The estimate of H. Davenport and H. Heilbronn [Proc. R. Soc. Lond., Ser. A 322, 405-420 (1971; Zbl 0212.081)] can then be applied.
For the entire collection see [Zbl 0801.00020].

Reviewer: D.R.Heath-Brown (Oxford)

MSC:

11G05	Elliptic curves over global fields
11R47	Other analytic theory

Keywords:

cubic twist; Selmer group; average behaviour; rank; elliptic curve

Citations:

Zbl 0212.081