High rank elliptic curves induced by rational Diophantine triples

Dujella, Andrej; Peral, Juan Carlos

doi:10.3336/gm.55.2.05

Cited by 6 publications

(4 citation statements)

References 21 publications

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“…We say that this elliptic curve is induced by the rational Diophantine triple {a, b, c}. The question of possible Mordell-Weil groups of such elliptic curve over Q, Q(t) and quadratic fields, was considered in several papers (see [1,5,8,13,19,20,21,22,23,32]). In particular, it is shown in [8] that all four torsion groups that are allowed by Mazur's theorem for elliptic curves with full 2-torsion, i.e.…”

Section: Introductionmentioning

confidence: 99%

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On elliptic curves induced by rational Diophantine quadruples

Dujella,

Soydan

2021

Preprint

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In this paper, we consider elliptic curves induced by rational Diophantine quadruples, i.e. sets of four nonzero rationals such that the product of any two of them plus 1 is a perfect square. We show that for each of the groups Z/2Z × Z/kZ for k = 2, 4, 6, 8, there are infinitely many rational Diophantine quadruples with the property that the induced elliptic curve has this torsion group. We also construct curves with moderately large rank in each of these four cases.

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Section: Introductionmentioning

confidence: 99%

“…Let us mention that the curves with the largest known rank over Q and torsion groups Z/2Z × Z/kZ for k = 4, 6 and 8 can be induced by rational Diophantine triples (see e.g. [22]). The details about the current rank records and the corresponding curves can be found at the web page [12].…”

Section: Introductionmentioning

confidence: 99%

On elliptic curves induced by rational Diophantine quadruples

Dujella,

Soydan

2021

Preprint

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“…Parametric formulas for Diophantine triples can be useful as a starting point in the construction of Diophantine sets and the corresponding elliptic curves of high rank. See Section 2 in [Duj09] for the first such application to rational Diophantine sextuples, and [DKP20], [DP20b] and [DP20a] for three interesting applications of Luka's formulas.…”

Section: Introductionmentioning

confidence: 99%

Diophantine triples and K3 surfaces

Kazalicki¹,

Naskręcki²

2021

Preprint

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A Diophantine m-tuple with elements in the field K is a set of m non-zero (distinct) elements of K with the property that the product of any two distinct elements is one less than a square in K. Let X : (x 2 − 1)(y 2 − 1)(z 2 − 1) = k 2 , be an affine variety over K. Its K-rational points parametrize Diophantine triples over K such that the product of the elements of the triple that corresponds to the point (x, y, z, k) ∈ X(K) is equal to k. We denote by X the projective closure of X and for a fixed k by X k a variety defined by the same equation as X.In this paper, we try to understand what can the geometry of varieties X k , X and X tell us about the arithmetic of Diophantine triples.First, we prove that the variety X is birational to P 3 which leads us to a new rational parametrization of the set of Diophantine triples.Next, specializing to finite fields, we find a Shioda-Inose structure of the K3 surface X k for a given k ∈ F × p in the prime field F p of odd characteristic, determined by an abelian surface which is a product of two elliptic curvesWe derive an explicit formula for N (p, k), the number of Diophantine triples over F p with the product of elements equal to k. Moreover, we show that the variety X admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on X over an arbitrary finite field F q . Using it we reprove the formula for the number of Diophantine triples over F q from [DK21].Curiously, from the interplay of the two (K3 and rational) fibrations of X, we derive the formula for the second moment of the elliptic surface E k (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating S 4 (Γ 0 (8)).Finally, in the Appendix, Luka Lasić defines circular Diophantine m-tuples, and describes the parametrization of these sets. For m = 3 this method provides an elegant parametrization of Diophantine triples.

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“…Moreover, it was shown that every elliptic curve with torsion group Z/2Z × Z/8Z is induced by a Diophantine triple (see also [4]). Questions about the ranks of elliptic curves induced by Diophantine triples were studied in several papers ( [1,7,8,10,12,18,19,20,21]). In particular, such curves were used for finding elliptic curves with the largest known rank over Q and Q(t) with torsion groups Z/2Z × Z/4Z ( [18,20]) and Z/2Z × Z/6Z ( [19]).…”

Section: Introductionmentioning

confidence: 99%

Rank zero elliptic curves induced by rational Diophantine triples

Dujella

Mikić

2020

Rad Hrvatske akademije znanosti i umjetnosti

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Rational Diophantine triples, i.e. rationals a, b, c with the property that ab+1, ac+1, bc+1 are perfect squares, are often used in construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.

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High rank elliptic curves induced by rational Diophantine triples

Cited by 6 publications

References 21 publications

On elliptic curves induced by rational Diophantine quadruples

On elliptic curves induced by rational Diophantine quadruples

Diophantine triples and K3 surfaces

Rank zero elliptic curves induced by rational Diophantine triples

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