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Fix an elliptic curve
and assume the Riemann Hypothesis for the
for every quadratic twist
. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of
. We derive from this an upper bound for the density of low-lying zeros of
that is compatible with the randommatrixmodels of Katz and Sarnak. We also show that for any unbounded increasing function
, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of
are less than
for almost all
We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any α > 1/6 (resp. α > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |Δ(E)| [Lt ]αNE12α is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and [ges ]2, respectively.
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