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Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. (English) Zbl 1317.11038

The authors have proved an asymptotic formula for the number of \(\mathrm{SL}_3(\mathbb{Z})\)-equivalence classes of integral ternary cubic forms which have bounded invariants.
As an application, they have obtained the average size of the 3-Selmer group of all elliptic curves when these are ordered by height. Specifically, they have shown that this is equal to 4. They also provide a proof of the fact that a positive proportion of all elliptic curves are of rank 0. Additionally, the authors have proven that a positive proportion of elliptic curves have analytic rank 0 as well.

MSC:

11E76 Forms of degree higher than two
11G05 Elliptic curves over global fields
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