On the conjecture of Birch and Swinnerton-Dyer

González-Avilés, Cristian D.

doi:10.1090/s0002-9947-97-01762-5

Cited by 14 publications

(6 citation statements)

References 21 publications

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“…In fact, we prove BSD(2) for many quadratic twists of general elliptic curves in [KL16]. Similar results for BSD(2) were previously only known for three quadratic twist families ( [Tia14] for the congruent number curve X 0 (32), [GA97], [CLTZ15] for X 0 (49) and [CCL16] for X 0 (36)).…”

Section: Thensupporting

confidence: 72%

Level Raising mod 2 and Obstruction to Rank Lowering

2017

International Mathematics Research Notices

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Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A obtained by Ribet's level raising theorem. For certain imaginary quadratic fields K satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of E and A over K have different parity, as predicted by the BSD conjecture. When the 2-Selmer rank of E is one, we further prove that the 2-Selmer rank of A can never be zero, revealing an obstruction to rank lowering which is unseen for p-Selmer groups for odd p.

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Section: Thensupporting

confidence: 72%

Level Raising mod 2 and Obstruction to Rank Lowering

2017

International Mathematics Research Notices

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“…Coates et al [3] [2], and Gonzalez-Avilés [16] 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. The deep results by Skinner-Urban [30] allow (in practice, see section 3 for instance) to establish the full version of the Birch and Swinnerton-Dyer conjecture for a large class of elliptic curves without CM.…”

Section: Introductionmentioning

confidence: 99%

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

Dąbrowski

Szymaszkiewicz

2017

Math. Comp.

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“…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]).…”

Section: Introductionmentioning

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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On the conjecture of Birch and Swinnerton-Dyer

Abstract: Abstract. In this paper we complete Rubin's partial verification of the conjecture for a large class of elliptic curves with complex multiplication by Q( √ −7).

Cited by 14 publications

References 21 publications

Level Raising mod 2 and Obstruction to Rank Lowering

Level Raising mod 2 and Obstruction to Rank Lowering

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

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

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