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Low-lying zeros of families of elliptic curves

Matthew P. Young
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Abstract

We study the low-lying zeros of various interesting families of elliptic curve L-functions. One application is an upper bound on the average analytic rank of the family of all elliptic curves. The upper bound obtained is less than two, which implies that a positive proportion of elliptic curves over the rationals have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. These results are conditional on the Generalized Riemann Hypothesis.

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Matthew P. Young. 2004-06-16. Low-lying zeros of families of elliptic curves. https://doi.org/10.1090/s0894-0347-05-00503-5

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