High-rank elliptic curves with torsion induced by Diophantine triples

Dujella, Andrej; Peral, Juan Carlos

doi:10.1112/s1461157014000023

Cited by 22 publications

(21 citation statements)

References 9 publications

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“…Curve with torsion Z/2Z×Z/4Z and rank 4 over Q(t). We just summarize the main results in our paper [11]. In that paper we have shown that the following triple…”

Section: 7mentioning

confidence: 71%

“…In our paper [11] we constructed a curve over Q(t) induced by Diophantine triples having rank 4 and torsion group is Z/2Z × Z/4Z. We also show an example of a curve with rank 9.…”

Section: Diophantine Quadruples and Elliptic Curvesmentioning

confidence: 99%

“…By comparing the j-invariant of the elliptic curve induced by this triple with the j-invariant induced by (11), we obtain the condition 2w 4 + 4w 3 + 2w 2 + 1 = square. The quartic curve defined by this equation is birationally equivalent to the elliptic curve Y 2 = X 3 − X 2 − 9X + 9 with rank 1 and a generator P = [0, 3].…”

Section: Torsion Z/2z × Z/8zmentioning

confidence: 99%

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Elliptic curves induced by Diophantine triples

Dujella

Peral

2018

RACSAM

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Given a Diophantine triple {c 1 (t), c 2 (t), c 3 (t)}, the elliptic curve over Q(t) induced by this triple, i.e.can have as torsion group one of the non cyclic groups in Mazur's theorem, i.e. Z/2Z × Z/2Z, Z/2Z × Z/4Z, Z/2Z × Z/6Z or Z/2Z × Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.2010 Mathematics Subject Classification. 11G05.

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“…Curve with torsion Z/2Z×Z/4Z and rank 4 over Q(t). We just summarize the main results in our paper [11]. In that paper we have shown that the following triple…”

Section: 7mentioning

confidence: 71%

“…In our paper [11] we constructed a curve over Q(t) induced by Diophantine triples having rank 4 and torsion group is Z/2Z × Z/4Z. We also show an example of a curve with rank 9.…”

Section: Diophantine Quadruples and Elliptic Curvesmentioning

confidence: 99%

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Elliptic curves induced by Diophantine triples

Dujella

Peral

2018

RACSAM

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“…and high rank see [14,9] and it can be obtained by taking σ 3 = 3/4 in (7). As before, we can transform the quartic to the elliptic curve Y 2 = X 3 + 1512X + 33588.…”

Section: Some Concluding Remarksmentioning

confidence: 99%

There Are Infinitely Many Rational Diophantine Sextuples

Dujella

Kazalicki

Mikić

et al. 2016

Int Math Res Notices

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A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples. set 1 16 , 33 16 , 17 4 , 105 16 found by Diophantus (see [4]). Euler found infinitely many rational Diophantine quintuples (see [20]), e.g. he was able to extend the integer Diophantine quadruple {1, 3, 8, 120} found by Fermat, to the rational quintuple 1, 3, 8, 120, 777480 8288641 . Let us note that Baker and Davenport [2] proved that Fermat's set cannot be extended to an integer Diophantine quintuple, while Dujella and Pethő [15] showed that there is no integer Diophantine quintuple which contains the pair {1, 3}. For results on the existence of infinitely many rational D(q)-quintuples, i.e. sets in which xy + q is always a square, for q = 1 see [12].

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“…Moreover, it was shown that every elliptic curve with torsion group Z/2Z × Z/8Z is induced by Diophantine triple (see also [7]). Questions about the ranks of elliptic curves induced by Diophantine triples was studied in several articles ( [1,10,12,16]). In particular, such curves were used for finding elliptic curves with the largest known rank with torsion group Z/2Z × Z/4Z ( [16]).…”

Section: Introductionmentioning

confidence: 99%

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

Dujella¹,

Bokun²,

Soldo³

2016

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The possible torsion groups of elliptic curves induced by Diophantine triples over quadratic fields, which do not appear over Q, are Z/2Z × Z/10Z, Z/2Z × Z/12Z and Z/4Z × Z/4Z. In this paper, we show that all these torsion groups indeed appear over some quadratic field. Moreover, we prove that there are infinitely many Diophantine triples over quadratic fields which induce elliptic curves with these torsion groups.

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High-rank elliptic curves with torsion induced by Diophantine triples

Cited by 22 publications

References 9 publications

Elliptic curves induced by Diophantine triples

Elliptic curves induced by Diophantine triples

There Are Infinitely Many Rational Diophantine Sextuples

On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields

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