Fermat’s Last Theorem and modular curves over real quadratic fields

Michaud‐Jacobs, Philippe

doi:10.4064/aa210812-2-4

Cited by 9 publications

(7 citation statements)

References 42 publications

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“…Remark 4. 16 It is perhaps worth noting at this point that the results of this section could be suitably extended to number fields of larger degree (we refer to [3] for a selection of such results). There are two main reasons we have chosen to focus on the case of quadratic fields.…”

Section: P Michaud-jacobsmentioning

confidence: 99%

“…However, it does not consider the case (y, y τ ) Fq = (∞, 0) Fq , which is certainly possible if q splits in K. Our next result, from the author's own work, addresses this case (for which we do not need to assume that p is unramified in K). Lemma 4.12 [16,Lemma 4.8] Let p and q be primes. Let y ∈ X 0 (p)(K).…”

Section: On Elliptic Curves With P-isogenies Over Quadratic Fields 957mentioning

confidence: 99%

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On elliptic curves with p-isogenies over quadratic fields

Michaud‐Jacobs

2022

Can. J. Math.-J. Can. Math.

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Let K be a number field. For which primes p does there exist an elliptic curve E/K admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

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Section: P Michaud-jacobsmentioning

confidence: 99%

Section: On Elliptic Curves With P-isogenies Over Quadratic Fields 957mentioning

confidence: 99%

On elliptic curves with p-isogenies over quadratic fields

Michaud‐Jacobs

2022

Can. J. Math.-J. Can. Math.

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“…We note that the idea of using the modular parametrization map to study points on modular curves is present in the author's work in [16, pp. 16–21].…”

Section: The Equation X2ℓ+y2m=z17$x^{2\ell }+Y^{2m}=z^{17}$mentioning

confidence: 99%

On some generalized Fermat equations of the form x2+y2n=zp$x^2+y^{2n} = z^p$

Michaud‐Jacobs

2022

Mathematika

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The primary aim of this paper is to study the generalized Fermat equationin coprime integers 𝑥, 𝑦, and 𝑧, where 𝑛 ⩾ 2 and 𝑝 is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a complete resolution of this equation in the case 𝑝 = 7, and obtain an asymptotic result for fixed 𝑝. Additionally, using similar techniques, we solve a second equation, namely, 𝑥 2𝓁 + 𝑦 2𝑚 = 𝑧 17 , for primes 𝓁, 𝑚 ≠ 5.

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“…There has been a lot of recent interest in computing low-degree points on modular curves, and in particular in computing quadratic points on the curves X 0 (N ). Computing such points gives much insight into the arithmetic of elliptic curves and has direct applications in the resolution of Diophantine equations (see [8, p. 888] or [10] for such examples).…”

Section: Introductionmentioning

confidence: 99%