Towards an ‘average’ Version of the Birch and Swinnerton-dyer Conjecture
نویسندگان
چکیده
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 [Mes] showed the first zero occurs by O(1/ log logNE), where NE is the conductor of E, though we expect the correct scale to study the zeros near the central point is the significantly smaller 1/ logNE . 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 Squarefree 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/ logNE (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.
منابع مشابه
On the elliptic curves of the form $ y^2=x^3-3px $
By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
متن کاملThe Birch-swinnerton-dyer Conjecture
We give a brief description of the Birch-Swinnerton-Dyer conjecture which is one of the seven Clay problems.
متن کاملNumerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecture
In the first part of the talk we describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson. Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an ...
متن کاملON RUBIN’S VARIANT OF THE p-ADIC BIRCH AND SWINNERTON-DYER CONJECTURE
We study Rubin’s variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable p-adic L-function that lie outside the range of p-adic interpolation.
متن کاملComputational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over Q of analytic ranks 0 and 1. We apply our techniques to show that if E is a non-CM elliptic curve over Q of conductor ≤ 1000 and rank 0 or 1, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the L-series is true for...
متن کامل