TY - JOUR

T1 - Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0

AU - Bhargava, Manjul

AU - Shankar, Arul

N1 - Publisher Copyright:
© 2015 Department of Mathematics, Princeton University.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - We prove an asymptotic formula for the number of SL3(Z)-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is equal to 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have nonvanishing L-function at s = 1. It follows that a positive proportion of all elliptic curves satisfy BSD.

AB - We prove an asymptotic formula for the number of SL3(Z)-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is equal to 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have nonvanishing L-function at s = 1. It follows that a positive proportion of all elliptic curves satisfy BSD.

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U2 - 10.4007/annals.2015.181.2.4

DO - 10.4007/annals.2015.181.2.4

M3 - Article

AN - SCOPUS:84912036290

VL - 181

SP - 587

EP - 621

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 2

ER -