Elliptic curves over real quadratic fields are modular

Freitas, Nuno; Hung, Bao V. Le; Siksek, Samir

doi:10.1007/s00222-014-0550-z

Cited by 98 publications

(172 citation statements)

References 75 publications

(149 reference statements)

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“…Since E a,b and F a,b are defined over a real quadratic field, they are modular by the main result of [27]. This completes Step 3 of the modular method.…”

Section: The Image Of Inertia Argumentmentioning

confidence: 64%

“…We note that modularity of E a,b is guaranteed by [27], hence completing Step 3 of the modular method.…”

Section: Proof Recall That As Elements Ofmentioning

confidence: 82%

“…At the time, the purpose of using Q-curves was to guarantee their modularity. It is now known that elliptic curves over real quadratic fields are modular (see [27]) and therefore we can work directly over Q( √ 5), largely simplifying the arguments.…”

Section: The Image Of Inertia Argumentmentioning

confidence: 99%

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A multi-Frey approach to Fermat equations of signature $(r,r,p)$

Billerey

Chen

Dieulefait

et al. 2019

Trans. Amer. Math. Soc.

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Abstract. In this paper, we give a resolution of the generalized Fermat equations x 5 + y 5 = 3z n and x 13 + y 13 = 3z n , for all integers n ≥ 2, and all integers n ≥ 2 which are not a power of 7, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents n.We also give a number of results for the equations x 5 + y 5 = dz n , where d = 1, 2, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at p modulo p.

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“…Since E a,b and F a,b are defined over a real quadratic field, they are modular by the main result of [27]. This completes Step 3 of the modular method.…”

Section: The Image Of Inertia Argumentmentioning

confidence: 64%

“…We note that modularity of E a,b is guaranteed by [27], hence completing Step 3 of the modular method.…”

Section: Proof Recall That As Elements Ofmentioning

confidence: 82%

See 1 more Smart Citation

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

Billerey

Chen

Dieulefait

et al. 2019

Trans. Amer. Math. Soc.

Self Cite

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show abstract

“…, C n and finite separable morphisms f i : C i → S whose ramification indices are coprime to the characteristic of k, it describes the arithmetic genus of the normalisation of C 1 × S · · · × S C n . We note that there has already been a significant application of this result; in [14], the authors prove that elliptic curves over real quadratic fields are modular, and in one of their steps they use our Theorem 7 (citing a preprint version of this article) in order to describe points on certain modular curves (see Sect. 16.1 of [14] for details).…”

Section: Introductionmentioning

confidence: 99%

“…We note that there has already been a significant application of this result; in [14], the authors prove that elliptic curves over real quadratic fields are modular, and in one of their steps they use our Theorem 7 (citing a preprint version of this article) in order to describe points on certain modular curves (see Sect. 16.1 of [14] for details). Theorem 11, summarised above, in Sect.…”

Section: Introductionmentioning

confidence: 99%

Finite Weil restriction of curves

Flynn

Testa

2014

Monatsh Math

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Given number fields L ⊃ K , smooth projective curves C defined over L and B defined over K , and a non-constant L-morphism h : C → B L , we denote by C h the curve defined over K whose K -rational points parametrize the L-rational points on C whose images under h are defined over K . We compute the geometric genus of the curve C h and give a criterion for the applicability of the Chabauty method to find the points of the curve C h . We provide a framework which includes as a special case that used in Elliptic Curve Chabauty techniques and their higher genus versions. The set C h (K ) can be infinite only when C has genus at most 1; we analyze completely the case when C has genus 1.

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Taking a Leading Role as a “First Mover” to Advance Materials Science and Technology at the Ulsan National Institute of Science & Technology (UNIST)

Cho

2019

Advanced Materials

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We present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N = 1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N = 1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy.

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Elliptic curves over real quadratic fields are modular

Abstract: Abstract. We prove that all elliptic curves defined over real quadratic fields are modular.

Cited by 98 publications

References 75 publications

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

A multi-Frey approach to Fermat equations of signature $(r,r,p)$

Finite Weil restriction of curves

Taking a Leading Role as a “First Mover” to Advance Materials Science and Technology at the Ulsan National Institute of Science & Technology (UNIST)

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