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- W4299682783 abstract "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 particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5." @default.
- W4299682783 created "2022-10-02" @default.
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- W4299682783 date "2010-06-04" @default.
- W4299682783 modified "2023-10-16" @default.
- W4299682783 title "Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves" @default.
- W4299682783 doi "https://doi.org/10.48550/arxiv.1006.1002" @default.
- W4299682783 hasPublicationYear "2010" @default.
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