Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0

@article{Bhargava2010TernaryCF,
  title={Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0},
  author={M. Bhargava and Arul Shankar},
  journal={arXiv: Number Theory},
  year={2010}
}
We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbb Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive… Expand
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