Erman IŞIK, Yasemin KARA, Ekin ÖZMAN KARAKURT

On ternary Diophantine equations of signature $ p,p,2 $ over number fields

Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Keywords:

Fermat equation generalized Fermat equation, S-units, modularity,

___

[1] Alvarado A, Koutsianas A, Malmskog B, Rasmussen C, Vincent C et al. A robust implementation for solving the S -unit equation and several applications. arXiv:1903.00977
[2] Anni S, Siksek S. Modular elliptic curves over real abelian fields and the generalized Fermat equation x 2+y 2m = z p . Algebra & Number Theory 2016; 6 (10): 1147-1172.
[3] Bennett MA. The equation x 2n + y 2n = z 5 . Journal de théorie des nombres de Bordeaux 2006; 2 (18) : 315-321.
[4] Bennett MA, Chen I. Multi-Frey Q-curves and the Diophantine equation a 2 + b 6 = c n . Algebra & Number Theory 2012; 4 (6): 707-730.
[5] Bennett MA, Chen I, Dahmen SR, Yazdani S. Generalized Fermat equations: a miscellany. International Journal of Number Theory 2015; 1 (11): 1-28.
[6] Bennett MA, Ellenberg JS, Ng NC. The diophantine equation A 4 + 2δB2 = C n . International Journal of Number Theory 2010; 2 (6): 311-338.
[7] Bennet M, Mihăilescu P, Siksek S. The generalized Fermat equation, in Open problems in mathematics 2016; Springer, pp. 173-205.
[8] Bennett MA, Skinner CM. Ternary Diophantine equations via Galois representations and modular forms. Canadian Journal of Mathematics 2004; 1 (56): 23-54.
[9] Billerey N, Chen I, Dieulefait L, Freitas N. A multi-Frey approach to Fermat equations of signature(r, r, p).Transactions of American Mathematical Society 2019; 12 (371): 8651-8677.
[10] Billerey N, Chen I, Dieulefait L, Freitas N. A result on the equation x p + y p = z r using Frey abelian varieties. Proceedings of the American Mathematical Society 2017; 10 (145), pp.4111-4117.
[11] Billerey N, Dieulefait L. Solving Fermat-type equations x 5 + x 5 = dzp . Mathematics of Computation 2010; 269 (79), pp.535-544.
[12] Blasius D. Elliptic curves. Hilbert modular forms and the Hodge conjecture, In: Hida H, Ramakrishnan D, Shahidi F (editors). Contributions to Automorphic Forms, Geometry, and Number Theory. Baltimore, MD, USA: Johns Hopkins University Press, pp. 83-103.
[13] Brown D. Primitive Integral Solutions to x 2 + y 3 = z 10 , International Mathematics Research Notices IMRN 2012 : 423-436.
[14] Bruin N. The Diophantine Equations x 2 ± y 4 = z 6 and x 2 + y 8 = z 3 . Compositio Mathematica 1999; 3 (118): 305-321.
[15] Bruin N. On powers as sums of two cubes. In: International Algorithmic Number Theory Symposium 2000; July, Springer, Berlin, Heidelberg, pp. 169-184.
[16] Bruin N. The primitive solutions to x 3 + y 9 = z 2 . J. Number Theory 2005; 1 (111) : 179-189.
[17] Chen I. On the equation a 2 + b 2p = c 5 , Acta Arithmetica 2010; 4 (143): 345-375.
[18] Chen I. On the equation s 2 + y 2p = α 3 . Mathematical Componrnts 2008; 77 : 1223-1227.
[19] Chen I, Siksek S. Perfect powers expressible as sums of two cubes. Journal of Algebra 2009; 3 (322) : 638-656.
[20] Cohen H. Advanced Topics in Computational Number Theory. New York, NY, USA: Springer, 1999.
[21] Dahmen S. Classical and modular methods applied to Diophantine equations. Utrecht University, 2008.
[22] Dahmen S. A refined modular approach to the Diophantine equation x 2 + y 2n = z 3 . International Journal Number Theory 7 2011; 5 : 1303-1316.
[23] Dahmen SR, Siksek S. Perfect powers expressible as sums of two fifth or seventh powers. arXiv preprint 2013; arXiv:1309.4030.
[24] Darmon H, Granville A. On the equations z m = F(x, y) and Axp+Byq = Czr . Bulletin of the London Mathematical Society 1995; 6 (27): 513-543.
[25] Darmon H, Merel L. Winding quotients and some variants of Fermat’s Last Theorem. Journal Reine Angew Mathematik 1997; 490: 81-100.
[26] Deconinck H. On the generalized Fermat equation over totally real fields. Acta Arithmetica 2016; 3 (173): 225-237.
[27] Derickx M, Najman F, Siksek S. Elliptic curves over totally real cubic fields are modular. arXiv:1901.03436v1
[28] Dieulefait L, Freitas N. Fermat-type equations of signature (13, 13, p) via Hilbert cuspforms. Mathematische Annalen 2013; 3 (357): 987-1004.
[29] Ellenberg JS. Galois Representations Attached to Q-Curves and the Generalized Fermat Equation A 4 + B 2 = C p . American journal of mathematics 2004; 763-787.
[30] Faltings G. Erratum: Finiteness theorems for abelian varieties over number fields. Inventiones Mathematicae 1984; 2 (75): 381.
[31] Freitas N, Le Hung BV, Siksek S. Elliptic curves over real quadratic field are modular. Inventiones Mathematicae 2015; 1 (201): 159-206.
[32] Freitas N, Siksek S. Criteria for Irreducibility of mod p Representations of Frey Curves. Journal de théorie des nombres de Bordeaux 2015; 1 (27): 67-76.
[33] Freitas N, Siksek S. The asymptotic Fermat’s last theorem for five-sixths of real quadratic fields. Compositio Mathematica 2015; 8 (18): 1395-1415.
[34] Freitas N, Siksek S. Criteria for irreducibility of mod p representations of Frey curves. Journal de Theorie des Nombres de Bordeaux 2015; 1 (27): 67-76.
[35] Jarvis F, Meekin P. The Fermat equation over Q( √ 2). Journal of Number Theory 2004; (109) : 182-196.
[36] Kara Y, Ozman E. Asymptotic Generalized Fermat’s Last Theorem over Number Fields. To appear in International Journal of Number Theory. doi: 10.1142/S1793042120500463
[37] Katz NM. Galois properties of torsion points on abelian varieties. Inventiones Mathematicae 1981; 3 (62): 481-502.
[38] Koutsianas A. Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction. Experimental Mathematics 2019; 1 (28): 1-15.
[39] Kraus A. Majorations effectives pour l’équation de Fermat Généralisée. Canadian Journal of Mathematics 1997; 49 : 1139-1161 (in French).
[40] Kraus A. Sur l’équation a 3 + b 3 = c p . Experimental Mathematics 1998; 7 (1) : 1-13.
[41] Poonen B. Some Diophantine equations of the form x n + y n = z m . Acta Arithmetica 1998; 86 : 193-205.
[42] Poonen B, Schaefer EF, Stoll M. Twists of X(7) and primitive solutions to x 2 + y 3 = z 7 . Duke Mathematical Journal 2007; 1 (137): 103-158.
[43] Şengün MH, Siksek S. On the asymptotic Fermat’s last theorem over number fields. Commentarii Mathematici Helvetici 2018; 2 (93): 359-375.
[44] Siksek S. On the Diophantine equation x 2 = y p + 2k z p . Journal de Théorie des Nombres de Bordeaux 2003; 15 : 839-846.
[45] Siksek S. Explicit Chabauty over number fields. Algebra & Number Theory, 7 (4): 765-793.
[46] Siksek S, Stoll M. Partial descent on hyperelliptic curves and the generalized Fermat equation x 3 + y 4 + z 5 = 0. Bulletin of the LMS 2012; 44 : 151-166
[47] Siksek S, Stoll M. The generalised Fermat equation x 2 + y 3 = z 15 . Archiv Mathematik (Basel) 2014; 5 (102): 411-421.
[48] Silverman JH. The arithmetic of elliptic curves. Graduate Texts in Mathematics 1986; vol. 106, Springer, Dordrecht.
[49] Silverman JH. Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 1994; vol. 151. Springer, New York.
[50] Taylor R, Wiles A. Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics 1995; 141: 553-572.
[51] Turcas G. On Fermat’s equation over some quadratic imaginary number fields. Research in Number Theory 2018; 4:24.
[52] Turcas G. On Serre’s modularity conjecture and Fermat’s equation over quadratic imaginary fields of class number one. arXiv:1908.11690v1
[53] Wiles A. Modular elliptic curves and Fermat’s last theorem. Annals of Mathematics 1995; 2 (141): 443-551.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv