The square-free sieve and the rank of elliptic curves (original) (raw)
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Experimental Mathematics, 2006
The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N , there is a weight-3 2 modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution.
Ranks of twists of elliptic curves and Hilbert’s tenth problem
Inventiones mathematicae, 2010
In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field. This material is based upon work supported by the National Science Foundation under grants DMS-0700580 and DMS-0757807.
Experimental Mathematics, 2006
The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N , there is a weight-3 2 modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution.
Rank Zero Quadratic Twists of Modular Elliptic Curves
1996
In (11) L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D 1 mod 4N such that ED, the D quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ranks, simply by the sign of functional equations, it is not expected that there will be infinitely many square-free D in every arithmetic progression r (mod t) where gcd(r,t) is square-free such that ED has rank zero. Given a square-free positive integer r, under mild conditions we show that there exists an integer tr and a positive integer N where tr r mod Q ◊ 2
|X| Values on Quadratic Twists of Elliptic Curves
2006
The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N, there is a weight- 3 modular form g associated with it such that the d-coefficient of g is related to the value at s =1 of the L-series of the (�d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (�d)-quadratic twist of E and analyze their distribution.
A Note on Higher Twists of Elliptic Curves
Glasgow Mathematical Journal, 2010
We show that for any pair of elliptic curves E 1 , E 2 over ޑ with jinvariant equal to 0, we can find a polynomial D ∈ [ޚu, v] such that the cubic twists of the curves E 1 , E 2 by D(u, v) have positive rank over (ޑu, v). We also prove that for any quadruple of pairwise distinct elliptic curves E i , i = 1, 2, 3, 4, with j-invariant j = 0, there exists a polynomial D ∈ [ޚu] such that the sextic twists of E i , i = 1, 2, 3, 4, by D(u) have positive rank. A similar result is proved for quadruplets of elliptic curves with j-invariant j = 1, 728.
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Rocky Mountain J. Math., 2014
Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.
Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor
arXiv (Cornell University), 2022
Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the 2-adic valuation of its modular degree. We show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with rational 2-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form y 2 = x 3 − dx, with d a biquadratefree integer.
Height estimates on cubic twists of the Fermat elliptic curve
Bulletin of the Australian Mathematical Society 09/2005; 72(02):177 - 186.
We give bounds for the canonical height of rational and integral points on cubic twists of the Fermat elliptic curve. As a corollary we prove that there is no integral arithmetic progression on certain curves in this family
Average ranks of elliptic curves: Tension betweendata and conjecture
Bulletin of The American Mathematical Society, 2007
Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control-thanks to ideas of Kolyvagin [Kol88]-the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.