Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

  title={Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves},
  author={M. Bhargava and Arul Shankar},
  journal={arXiv: Number Theory},
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general, and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by their heights, is bounded. In… Expand

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