Low-lying zeros of families of elliptic curves

Details Low-lying zeros of families of elliptic curves Low-lying zeros of families of elliptic curves

2004-06-16

arxiv.org/abs/math/0406330v3

Mathematics

Number Theory

v2: Enhanced exposition, 56 pages. v3: One reference added and one sentence changed in the paragraph following Corollary 3.4

Scientific paper

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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