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

@article{Bhargava2010BinaryQF,
  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},
  year={2010}
}
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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References

SHOWING 1-10 OF 76 REFERENCES
Counting Elliptic Surfaces over Finite Fields
We count the number of isomorphism classes of elliptic curves of given height d over the field of rational functions in one variable over the finite field of q elements. We also estimate the numberExpand
The average analytic rank of elliptic curves
All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curvesExpand
REDUCTION OF BINARY CUBIC AND QUARTIC FORMS
A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in theExpand
The density of discriminants of quartic rings and fields
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is aExpand
Finiteness Theorems for Binary Forms with Given Discriminant
1. Classical invariant theory is concerned with the action of linear groups on spaces of algebraic forms and the algebraic invariants under such actions; in this paper we are concerned with one ofExpand
Higher composition laws III: The parametrization of quartic rings
In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields couldExpand
On the equivalence of binary quartics
TLDR
An extension and correction to a result stated in the first author's paper, concerning the equivalence of binary quartics, and gives an alternative criterion for equivalence. Expand
Mass Formulas for Local Galois Representations (with an Appendix by Daniel Gulotta)
Bhargava has given a formula, derived from a formula of Serre, computing a certain count of extensions of a local field, weighted by conductor and by number of automorphisms. We interpret this resultExpand
Average Ranks of Elliptic Curves: Tension between Data and Conjecture
Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are toExpand
Algorithms for Modular Elliptic Curves
This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation. It is in three parts. First, the author describes inExpand
...
1
2
3
4
5
...