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We prove a theorem giving the asymptotic number of binary quartic formshaving bounded invariants; this extends, to the quartic case, the classicalresults of Gauss and Davenport in the quadratic and cubic cases, respectively.Our techniques are quite general, and may be applied to counting integralorbits in other representations of algebraic groups.  We use these counting results to prove that the average rank of ellipticcurves over $\mathbb{Q}$, when ordered by their heights, is bounded. Inparticular, we show that when elliptic curves are ordered by height, the meansize of the 2-Selmer group is 3. This implies that the limsup of the averagerank of elliptic curves is at most 1.5.

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