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 occurswhere 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.

Moments of the Rank of Elliptic Curves

“…If instead of sending ǫ to zero we kept ǫ fixed, we are asking about the number of normalized zeros in a given neighborhood. The answer here can also be predicted from the Katz-Sarnak conjectures; using the one-level density Goes and Miller [12] have recently obtained explicit results for one-parameter families. Their calculations are similar to those by Mestre [28] and Hughes-Rudnick [15].…”

“…Since (1.6) is quasi-minimal for E, up to a bounded power of 2 and 3, the Weierstrass equation Denote by E A,B the Weierstrass equation y 2 = x 3 + Ax + B so that 4A 3 + 27B 2 = 0, and such that there exists no prime p with p 4 |A and p 6 |B. The latter condition implies that the discriminant of this equation differs from the minimal discriminant by at most 6 12 . Also, as H(E A,B ) := max(|A| 1/3 , |B| 1/2 ) goes to infinity, we capture all elliptic curves over Q.…”

A random matrix model for elliptic curveL-functions of finite conductor

“…There has been much success modelling families of L-functions with matrices from the classical compact groups [7,8,[10][11][12][13][14][15][16][17][18][19]21,22,[25][26][27][28][29][30][31][32][33][34][35][39][40][41][42][43]47,[52][53][54][55]57,58], including families of elliptic curve L-functions [13,14,37,38,48,49,62]. This work strongly supports the Katz-Sarnak philosophy that, in the correct asymptotic limit, matrices from the classical compact groups accurately model the zero statistics of L-functions.…”

Analytic properties of the Hasse–Weil L-function

“…Recently there are a number of publications relating to the Hasse-Weil L-functionss (e.g. [1,5,9]). Although L.E; s/ is introduced in many standard textbooks (e.g.…”