On Tate-Shafarevich groups of some elliptic curves

Lemmermeyer, Franz

doi:10.1515/9783110801958.277

Cited by 11 publications

(22 citation statements)

References 16 publications

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“…We will use the notation from [10]. The curve E is 2-isogenous to E : y 2 = x(x 2 + pq 2 ), and it is easy to see that …”

Section: Modern Interpretationmentioning

confidence: 99%

A note on pépin’s counter examples to the hasse principle for curves of genus 1

Lemmermeyer

1999

Abh.Math.Semin.Univ.Hambg.

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In a series of articles published in the C.R. Paris more than a century ago, T. Pépin announced a list of "theorems" concerning the solvability of diophantine equations of the type ax 4 + by 4 = z 2 . In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.

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“…We will use the notation from [10]. The curve E is 2-isogenous to E : y 2 = x(x 2 + pq 2 ), and it is easy to see that …”

Section: Modern Interpretationmentioning

confidence: 99%

A note on pépin’s counter examples to the hasse principle for curves of genus 1

Lemmermeyer

1999

Abh.Math.Semin.Univ.Hambg.

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“…It is assumed that the integers N are greater than 1 and coprime to 6. Lemmermeyer [4] remarked that there are Proth versions in which only part of N ± 1 needs to be factored and commented that in the same way Gross [2] gave an elliptic curve 'Lucas-Lehmer' test, the Lucas-Lehmer test itself may be proved with the Pell conic C : X 2 − 12Y 2 = 4 and the point (4,1). This method is extended here to more general primality proving.…”

Section: Introductionmentioning

confidence: 99%

“…Like elliptic curves, there is a group law on the Pell conics [4]. These are affine curves of genus 0 of the form C : X 2 − ΔY 2 = 4 where Δ is a fundamental discriminant.…”

Section: Introductionmentioning

confidence: 99%

“…(2) Lemmermeyer [4] considered the arithmetic of Pell conics indicating many interesting similarities with elliptic curves. The following theorem appears in [4] in a more general form. We use Proof.…”

Section: Introductionmentioning

confidence: 99%

“…Lemmermeyer [4] discussed primality proving using Pell conics, giving Theorem 1.5 as an analogue of Lucas' theorem. It is assumed that the integers N are greater than 1 and coprime to 6.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Lucas-Lehmer tests using Pell conics

Hambleton

2012

Proc. Amer. Math. Soc.

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On Elliptic Curves with Large Tate–Shafarevich Groups

Atake

2001

Journal of Number Theory

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The Tate-Shafarevich group X(E/K) of an elliptic curve E over a number field K is defined as X(E/K) := Ker H 1 (K, E(K)) −→ v H 1 (K v , E(K v) , where v runs over all (archimedean and non-archimedean) primes of K. This X(E/K) describes the failure of "Hasse principle" for the torsors of E/K. It is conjectured that X(E/K) is finite (still unknown in general). The following question is very natural, but we don't know the answer. Question 1. For each prime p, does there exist an elliptic curve E over Q such that the p-torsion subgroup X(E/Q)[p] of X(E/Q) is nonzero? We can give examples of such elliptic curves for small primes. • E : y 2 = x 3 + 17x ⇒ X(E/Q)[2] = 0. (Lind, Reichardt, '40s) • E : y 2 = x 3 − 24300 ⇒ X(E/Q)[3] = 0. (Selmer, 1951) • E : y 2 + xy = x 3 − 3301465x − 2309192023 ⇒ X(E/Q)[5] = 0. • E : y 2 + xy = x 3 − 3674496x − 2711401518 ⇒ X(E/Q)[7] = 0. • E : y 2 = x 3 +x 2 −21477749985x−1211529110734587 ⇒ X(E/Q)[13] = 0. Remark. One can verify the above assertions for p = 5 and 7 by a p-descent argument (cf. [Be], [Fi]) or by a result of Cassels [Ca2] together with the fact that rankE(Q) = 0. The second argument also works for p = 13 (cf. [Ma]). • E : y 2 = x 3 − 20675209x ⇒ X(E/Q)[11] = 0. • E : y 2 = x 3 − 239228089x ⇒ X(E/Q)[17] = 0. • E : y 2 = x 3 − 258904415517049x ⇒ X(E/Q)[211] = 0. Remark. These curves have complex multiplication by Z[ √ −1]. For such CM curves, Rubin [Ru] proved the full Birch and Swinnerton-Dyer conjecture (mod-ulo 2-parts). The above assertions is verified by using this fact and a result of Tunnell [Tu]. Another computation can be found in [Ro]. Moreover the following question has already been solved affirmatively for p ≤ 5.

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On Tate-Shafarevich groups of some elliptic curves

Abstract: Generalizing results of Stroeker and Top we show that the 2-ranks of the Tate-Shafarevich groups of the elliptic curves y 2 = (x + k)(x 2 + k 2 ) can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves.

Cited by 11 publications

References 16 publications

A note on pépin’s counter examples to the hasse principle for curves of genus 1

A note on pépin’s counter examples to the hasse principle for curves of genus 1

Generalized Lucas-Lehmer tests using Pell conics

On Elliptic Curves with Large Tate–Shafarevich Groups

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