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

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

2010-06-04

arxiv.org/abs/1006.1002v2

Mathematics

Number Theory

50 pages; abstract fixed; file unchanged

Scientific paper

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 integer orbits in other representations of algebraic groups. As an example, we illustrate the method on a particularly rich nonreductive representation having dimension 12. We use these counting results to deduce a number of arithmetic consequences. First, we determine the mean number of 2-torsion elements in the class groups of monogenic maximal cubic orders. We similarly determine the mean number of 2-torsion elements in the class groups of maximal cubic orders having a monogenic subring of bounded index. Surprisingly, we find that these mean values are different! This demonstrates that, on average, the monogenicity of a ring has a direct altering effect on the behavior of the class group. Finally, we utilize all the above results to prove that the average rank of elliptic curves, when ordered by their heights, is bounded. In particular, we prove that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the average rank of elliptic curves is at most 1.5.

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