TY - JOUR

T1 - Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

AU - Bhargava, Manjul

AU - Shankar, Arul

N1 - Publisher Copyright:
© 2015 Department of Mathematics, Princeton University.

PY - 2015

Y1 - 2015

N2 - We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, theclassical 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 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.

AB - We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, theclassical 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 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.

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U2 - 10.4007/annals.2015.181.1.3

DO - 10.4007/annals.2015.181.1.3

M3 - Article

AN - SCOPUS:84911410524

SN - 0003-486X

VL - 181

SP - 191

EP - 242

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -