Towards an ‘average’ version of the Birch and Swinnerton-Dyer conjecture

Abstract

Text The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell–Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L ( s , E ) . Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a family. Mestre (1986) [Mes] showed the first zero occurs by O ( 1 / log log N E ) , where N E is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1 / log N E . We significantly improve on Mestre's result by averaging over a one-parameter family of elliptic curves E over Q ( T ) . We assume GRH, Tate's conjecture if E is not a rational surface, and either the ABC or the Square-Free Sieve Conjecture if the discriminant has an irreducible polynomial factor of degree at least 4. We find non-trivial upper and lower bounds for the average number of normalized zeros in intervals on the order of 1 / log N E (which is the expected scale). Our results may be interpreted as providing further evidence in support of the Birch and Swinnerton-Dyer conjecture, as well as the Katz–Sarnak density conjecture from random matrix theory (as the number of zeros predicted by random matrix theory lies between our upper and lower bounds). These methods may be applied to additional families of L-functions. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3EVYPNi_LG0.