Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture - Mathematics > Number TheoryReport as inadecuate




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Abstract: The Birch and Swinnerton-Dyer conjecture states that the rank of theMordell-Weil group of an elliptic curve E equals the order of vanishing at thecentral point of the associated L-function Ls,E. Previous investigations havefocused on bounding how far we must go above the central point to be assured offinding a zero, bounding the rank of a fixed curve or on bounding the averagerank in a family. Mestre showed the first zero occurs by O1-loglogN E,where N E is the conductor of E, though we expect the correct scale to studythe zeros near the central point is the significantly smaller 1-logN E. Wesignificantly improve on Mestre-s result by averaging over a one-parameterfamily of elliptic curves, obtaining non-trivial upper and lower bounds for theaverage number of normalized zeros in intervals on the order of 1-logN Ewhich is the expected scale. Our results may be interpreted as providingfurther evidence in support of the Birch and Swinnerton-Dyer conjecture, aswell as the Katz-Sarnak density conjecture from random matrix theory as thenumber of zeros predicted by random matrix theory lies between our upper andlower bounds. These methods may be applied to additional families ofL-functions.



Author: John Goes, Steven J Miller

Source: https://arxiv.org/







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