Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves

Radziwiłł, Maksym; Soundararajan, K.

doi:10.1007/s00222-015-0582-z

Cited by 55 publications

(43 citation statements)

References 33 publications

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“…The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [6], Delaunay [11][12], Watkins [33], Radziwi l l-Soundararajan [24] (see also the papers [9] [7] [8] and references therein) substantially extend the systematic tables given by Cremona.…”

Section: Introductionmentioning

confidence: 66%

“…It is an interesting question to find results (or at least a conjecture) on distribution of the order of the Tate-Shafarevich group for rank zero Neumann-Setzer type elliptic curves E 1 (u) and E 2 (u). It turns out that the values of log(|X(E i (u))|/ |u|) are the natural ones to consider (compare Conjecture 1 in [24], and numerical experiments in [7] [8]). Below we create histograms from the data log(|X(E i (u))|/ |u|) − µ i log log |u| / σ 2 i log log |u| : |u| ∈ W , where µ 1 = − 1 2 , µ 2 = − 1 2 − log 2, σ 2 1 = 1, and σ 2 2 = 1 + (log 2) 2 (here we use Lemma 1(iii) above, and Lemma 4 in [24]).…”

Section: Distribution Of |X(e(u))|mentioning

confidence: 99%

“…It turns out that the values of log(|X(E i (u))|/ |u|) are the natural ones to consider (compare Conjecture 1 in [24], and numerical experiments in [7] [8]). Below we create histograms from the data log(|X(E i (u))|/ |u|) − µ i log log |u| / σ 2 i log log |u| : |u| ∈ W , where µ 1 = − 1 2 , µ 2 = − 1 2 − log 2, σ 2 1 = 1, and σ 2 2 = 1 + (log 2) 2 (here we use Lemma 1(iii) above, and Lemma 4 in [24]). Our data suggest that the values log(|X(E i (u))|/ |u|) also follow an approximate normal distribution.…”

Section: Distribution Of |X(e(u))|mentioning

confidence: 99%

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Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves

Dąbrowski

Szymaszkiewicz

2017

Math. Comp.

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Section: Introductionmentioning

confidence: 66%

Section: Distribution Of |X(e(u))|mentioning

confidence: 99%

Section: Distribution Of |X(e(u))|mentioning

confidence: 99%

See 1 more Smart Citation

Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves

Dąbrowski

Szymaszkiewicz

2017

Math. Comp.

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“…It is an interesting question to find results (or at least a conjecture) on distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve over Q. It turns out that the values of log(|X(E d )|/ √ d) are the natural ones to consider (compare Conjecture 1 in [25], and numerical experiments in [9]). Below we create histograms from the data…”

Section: Distribution Of |X(e D )|mentioning

confidence: 99%

Behaviour of the Order of Tate–Shafarevich Groups for the Quadratic Twists of $$X_0(49)$$

Dąbrowski

Jędrzejak

Szymaszkiewicz

2016

Springer Proceedings in Mathematics &Amp; Statistics

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We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve E over Q and positive integer k, an asymptotic formula for the number of quadratic twists E d , d positive square-free integers less than X, with finite group E d (Q) and |X(E d (Q))| = k 2 . This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve X 0 (49). In section 8 we exhibit 88 examples of rank zero elliptic curves with |X(E)| > 63408 2 , which was the largest previously known value for any explicit curve. Our record is an elliptic curve E with |X(E)| = 1029212 2 .Birch and Swinnerton-Dyer conjecture relates the arithmetic data of E to the behaviour of L(E, s) at s = 1.Conjecture 1 (Birch and Swinnerton-Dyer) (i) L-function L(E, s) has a zero of order r = rank E(Q) at s = 1,If X(E) is finite, the work of Cassels and Tate shows that its order must be a square.The first general result in the direction of this conjecture was proven for elliptic curves E with complex multiplication by Coates and Wiles in 1976 [6], who showed that if L(E, 1) = 0, then the group E(Q) is finite. Gross and Zagier [18] showed that if L(E, s) has a first-order zero at s = 1, then E has a rational point of infinite order. Rubin [26] proves that if E has complex multiplication and L(E, 1) = 0, then X(E) is finite. Let g E be the rank of E(Q) and let r E the order of the zero of L(E, s) at s = 1. Then Kolyvagin [20] proved that, if r E ≤ 1, then r E = g E and X(E) is finite. Very recently, Bhargava, Skinner and Zhang [1] proved that at least 66.48% of all elliptic curves over Q, when ordered by height, satisfy the weak form of the Birch and Swinnerton-Dyer conjecture, and have finite Tate-Shafarevich group.When E has complex multiplication by the ring of integers of an imaginary quadratic field K and L(E, 1) is non-zero, the p-part of the Birch and Swinnerton-Dyer conjecture has been established by Rubin [27] for all primes p which do not divide the order of the group of roots of unity of K. Coates et al. [5] [4], and Gonzalez-Avilés [17] showed that there is a large class of explicit quadratic twists of X 0 (49) whose complex L-series does not vanish at s = 1, and for which the full Birch and Swinnerton-Dyer conjecture is valid (covering the case p = 2 when K = Q( √ −7)). The deep results by Skinner-Urban ([29], Theorem 2) (see also Theorem 7 in section 8.4 below) allow, in specific cases (still assuming L(E, 1) is non-zero), to establish p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves without complex multiplication for all odd primes p (see examples in section 8.4 below, and section 3 in [10]). The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [7], Delaunay [12][13], Watkins [31], Radziwi l l-Soundararajan [25] (see also the papers [11][10] [9], and references therein) substantially extend the systematic tables given by...

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“…Further, by using the result, one will be able to obtain some consequences on the size of the Tate-Shafarevich group assuming the Birch and Swinnerton-Dyer conjecture for rank zero quadratic twists of a given elliptic curve, and the size of the Fourier coefficients of half-integral weight modular forms, using the work of Waldspurger. The corresponding results on quadratic twists of Elliptic curves have been proved in the recent paper of Radziwiłł and Soundararajan [14].…”

Section: Introductionmentioning

confidence: 91%