Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Dieulefait, Luis; Urroz, Jorge Jiménez

doi:10.1515/crelle.2009.064

Cited by 18 publications

(35 citation statements)

References 22 publications

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“…In general, for forms of higher degree, the problem rapidly becomes more complicated -to illustrate its difficulty, it is worthwhile noting that the affirmative answer to this question, in case F (x, y) = xy(x + y) or x(x 2 − y 2 ), and z = 0, ±1, is equivalent to the asymptotic version of Wiles' theorem [58], née Fermat's Last Theorem, and to a special case of a result of Darmon and Merel [9], respectively. For irreducible forms (over Q[x, y]), the only results in the literature answering our question affirmatively are for certain diagonal quartic forms, such as F (x, y) = x 4 + y 4 (together with "covering" forms of higher degree), due to Ellenberg [16] and to Dieulefait and Jiménez Urroz [12]) (see also [2]). In particular, there are hitherto no primitive, irreducible cubic forms for which the above question has been addressed.…”

Section: Introductionmentioning

confidence: 72%

“…The first is to solve the equations, as in Tzanakis and de Weger [53], through a combination of lower bounds in linear forms in complex and p-adic logarithms, with techniques from computational Diophantine approximation. A second method is to appeal to the syzygy (12), which shifts the problem to one of determining the S-integral points on a collection of (Mordell) elliptic curves. Since the number of such curves is exponential in the cardinality of S F , in many situations this method appears impractical.…”

Section: Examples and Computationsmentioning

confidence: 99%

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Klein forms and the generalized superelliptic equation

Bennett

Dahmen

2013

Ann. Math.

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If F (x, y) ∈ Z[x, y] is an irreducible binary form of degree k ≥ 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equationhas, given an integer l ≥ max{2, 7 − k}, at most finitely many solutions in coprime integers x, y and z. In this paper, for large classes of forms of degree k = 3, 4, 6 and 12 (including, heuristically, "most" cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.

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Section: Introductionmentioning

confidence: 72%

Section: Examples and Computationsmentioning

confidence: 99%

Klein forms and the generalized superelliptic equation

Bennett

Dahmen

2013

Ann. Math.

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“…The authors have subsequently learned that a similar technique for finding γ also appeared in [Dieulefait and Urroz 2009] (where K β is polyquadratic).…”

Section: Review Of ‫-ޑ‬Curves and Their Attached Galois Representationsmentioning

confidence: 99%

Multi-Frey ℚ-curves and the Diophantine equationa²+b⁶=cⁿ

Bennett

Chen

2012

ANT

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Multi-Frey= c n has no nontrivial positive integer solutions with (a, b) = 1 via a combination of techniques based upon the modularity of Galois representations attached to certain ‫-ޑ‬curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek. IntroductionFollowing the proof of Fermat's last theorem [Wiles 1995], there has developed an extensive literature on connections between the arithmetic of modular abelian varieties and classical Diophantine problems, much of it devoted to solving generalized Fermat equations of the shapein coprime integers a, b, and c, and positive integers p, q, and r . That the number of such solutions (a, b, c) is finite, for a fixed triple ( p, q, r ), is a consequence of [Darmon and Granville 1995]. It has been conjectured that there are in fact at most finitely many such solutions, even when we allow the triples ( p, q, r ) to vary, provided we count solutions corresponding to 1 p + 2 3 = 3 2 only once. Being extremely optimistic, one might even believe that the known solutions constitute a complete list, namely (a, b, c, p, q, r ) corresponding to 1 p + 2 3 = 3 2 , for p ≥ 7, and to nine other identities (see [Darmon and Granville 1995;Beukers 1998]

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“…(iv) Dieulefait and Urroz [15] used the method of Galois representations attached to Q-curves to solve the Diophantine equation 3x 2 + y 4 = z n . The authors suggest that their method can be applied to solve this type of equations with 3 replaced by other values of a.…”

Section: Introductionmentioning

confidence: 99%

On a class of Lebesgue-Ljunggren-Nagell type equations

Dąbrowski

Günhan

Soydan

2020

Journal of Number Theory

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Given odd, coprime integers a, b (a > 0), we consider the Diophantine equation ax 2 + b 2l = 4y n , x, y ∈ Z, l ∈ N, n odd prime, gcd(x, y) = 1. We completely solve the above Diophantine equation for a ∈ {7, 11, 19, 43, 67, 163}, and b a power of an odd prime, under the conditions 2 n−1 b l ≡ ±1(mod a) and gcd(n, b) = 1. For other square-free integers a > 3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd(x, y) = 1), l ∈ N and all odd primes n > 3, satisfying 2 n−1 b l ≡ ±1(mod a), gcd(n, b) = 1, and gcd(n, h(−a)) = 1, where h(−a) denotes the class number of the imaginary quadratic field Q( √ −a).

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Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Cited by 18 publications

References 22 publications

Klein forms and the generalized superelliptic equation

Klein forms and the generalized superelliptic equation

Multi-Frey ℚ-curves and the Diophantine equationa²+b⁶=cⁿ

On a class of Lebesgue-Ljunggren-Nagell type equations

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Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields

Cited by 18 publications

References 22 publications

Klein forms and the generalized superelliptic equation

Klein forms and the generalized superelliptic equation

Multi-Frey ℚ-curves and the Diophantine equationa2+b6=cn

On a class of Lebesgue-Ljunggren-Nagell type equations

Contact Info

Product

Resources

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Multi-Frey ℚ-curves and the Diophantine equationa²+b⁶=cⁿ