On the Distribution of Analytic Values on Quadratic Twists 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.
|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.
Distribution of Selmer ranks of quadratic twists of elliptic curves
2016
We study the distribution of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We first prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. Under the assumption that Gal(K(E[2])/K) = S 3 we also show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on the "parity fraction" alluded to above, of a certain Markov Process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual Fp-representations of the absolute Galois group of K by characters of order p. Contents 6. Parity disparity (p = 2) 7. Parity (p > 2) 8. Changing Selmer ranks 9. Local and global characters, when the image of Galois is big 10. Probability distributions 11. Rank densities References
Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions
Ranks of Elliptic Curves and Random Matrix Theory
We examine the number of vanishings of quadratic twists of the Lfunction associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.
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
Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor
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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.
On the Vanishing of TwistedL-Functions of Elliptic Curves
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Let E be an elliptic curve over Q with L-function LE(s). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted Lfunctions LE(1, χ), as χ runs over the Dirichlet characters of order 3 (cubic twists). We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values LE(1, χ) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of LE(s).