Recipes to Fermat-type equations of the form $$x^r + y^r =Cz^p$$ x r + y r = C z p

Freitas, Nuno

doi:10.1007/s00209-014-1384-5

Cited by 16 publications

(33 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…From Lemma 9 we know that F a,b is semistable at all primes dividing 3 in K. Thus, from [26,Theorem 6.3], it follows that F a,b is modular.…”

Section: Using the Equalities Abmentioning

confidence: 93%

“…More generally, in the work of the last author [26], for each r ≥ 5, several Frey elliptic curves defined over real subfields of Q(ζ r ) are attached to equation (1.2). Our proofs of Theorems 1 and 2 build on these previous works and are made possible by introducing new multi-Frey techniques.…”

mentioning

confidence: 99%

“…The modular method over the real cubic subfield of Q(ζ 13 ). In [26], several Frey elliptic curves are attached to equation (1.2). In particular, for r = 13 one of them is E a,b from the previous section; in this section we will use another Frey elliptic curve adapted from a construction in loc.…”

mentioning

confidence: 99%

“…This curve is (up to a rational isomorphism) the quadratic twist by 13 of the curve defined by equation (13) with (k 1 , k 2 ) = (1, 5) in [26].…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Billerey

Chen

Dieulefait

et al. 2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…From Lemma 9 we know that F a,b is semistable at all primes dividing 3 in K. Thus, from [26,Theorem 6.3], it follows that F a,b is modular.…”

Section: Using the Equalities Abmentioning

confidence: 93%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…This curve is (up to a rational isomorphism) the quadratic twist by 13 of the curve defined by equation (13) with (k 1 , k 2 ) = (1, 5) in [26].…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Billerey

Chen

Dieulefait

et al. 2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Attention is now shifting towards Diophantine equations where the Frey elliptic curves are defined over totally real fields (for example [2], [6], [7], [8]). One now finds in the literature some of the necessary modularity (e.g.…”

Section: Introductionmentioning

confidence: 99%

Criteria for Irreducibility of mod p Representations of Frey Curves

Freitas¹,

Siksek²

2015

J. Théor. Nombres Bordeaux

Self Cite

View full text Add to dashboard Cite

Soit K un corps galoisien totalement réel, et soit E un ensemble de courbes elliptiques sur K. Nous donnons des conditions suffisantes pour l'existence d'un ensemble calculable de premiers rationnels P tels que, pour p / ∈ P et E ∈ E, la représentation Gal(K/K) → Aut(E[p]) soit irréductible. Nos conditions sont en général satisfaites par les courbes de Frey associées à des solutions d'équations diophantiennes; dans ce contexte, l'irréductibilité de la mod p représentation est une hypothèse requise pour l'application des théorèmes d'abaissement du niveau. Comme illustration de notre approche, nous avons amélioré le résultat de [6] pour les équations de Fermat de signature (13, 13, p).

show abstract