A conditional determination of the average rank of elliptic curves
نویسندگان
چکیده
منابع مشابه
On the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
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Let E=Q(T) be a one-parameter family of elliptic curves. Assuming various standard conjectures, we give an upper bound for the average rank of the bers E t (Q) with t 2 Z, improving earlier estimates of Fouvry-Pomykala and Michel. We also show how certain assumptions about the distribution of zeros of L-series might help explain the experimentally observed fact that the average rank of the bers...
متن کاملon the rank of certain parametrized elliptic curves
in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.
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Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the Lfunction L(ED, s) for every quadratic twist ED of E by D ∈ Z. 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 ED. We derive from this an upper bound for the density of low-lying zeros of L(ED, s) which is compatible with the...
متن کاملOn the Rank of Elliptic Curves
Notation. K will always denote a number eld. xI. Ranks of Families of Elliptic Curves For our purposes, a family of elliptic curves will be given by an equation E : y 2 = x 3 + A(T)x + B(T); A(T); B(T) 2 KT]; (T) = 4A(T) 3 + 27B(T) 2 6 = 0: We will always assume that E is non-split (i.e., is not obtained by base extension from a curve deened over K). The rank Generic Rank = Rank E(K(T)) is the ...
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ژورنال
عنوان ژورنال: Journal of the London Mathematical Society
سال: 2016
ISSN: 0024-6107,1469-7750
DOI: 10.1112/jlms/jdw058