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Level Lowering Modulo Prime Powers and Twisted Fermat Equations

Published online by Cambridge University Press:  20 November 2018

Sander R. Dahmen  and
Soroosh Yazdani
Sander R. Dahmen
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2 email: dahmen@math.ubc.ca
Soroosh Yazdani
Affiliation:
Department of Mathematics and Statistics, McMaster University, West Hamilton, ON, L8S 4K1 email: syazdani@math.mcmaster.ca
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Abstract

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We discuss a clean level lowering theorem modulo prime powers for weight 2 cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $a{{x}^{n}}\,+\,b{{y}^{n\,}}\,+\,c{{z}^{n}}\,=\,0$.

Keywords

11D4111F3311F1111F8011G05modular formslevel loweringDiophantine equations
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 282 - 300
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Darmon, H., Diamond, F., and Taylor, R., Fermat's last theorem. In: Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 1–154.Google Scholar
[2] Darnell, M., Holden, C., Kane, B., Weinstein, J., and Yazdani, S., MSRI modular forms summer workshop. August 2006, http://www.math.ubc.ca/_syazdani/MSRI archive Google Scholar
[3] Diamond, F., On deformation rings and Hecke rings. Ann. of Math. (2) 144(1996), no. 1, 137–166. http://dx.doi.org/10.2307/2118586 Google Scholar
[4] Dieulefait, L. and Taixési Ventosa, X., Congruences between modular forms and lowering the level mod ln. J. Théor. Nombres Bordeaux 21(2009), no. 1, 109–118.Google Scholar
[5] Halberstadt, E. and Kraus, A., Courbes de Fermat: résultats et problèmes. J. Reine Angew. Math. 548(2002), 167–234. http://dx.doi.org/10.1515/crll.2002.058 Google Scholar
[6] Kraus, A., Majorations effectives pour l’équation de Fermat généralisée. Canad. J. Math. 49(1997), no. 6, 1139–1161. http://dx.doi.org/10.4153/CJM-1997-056-2 Google Scholar
[7] Kraus, A., Sur l’équation a3 + b3 = cp. Experiment. Math. 7(1998), no. 1, 1–13.Google Scholar
[8] Ribet, K. A., On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(1990), no. 2, 431–476. http://dx.doi.org/10.1007/BF01231195 Google Scholar
[9] Ribet, K. A., Report on mod l representations of Gal(Q/Q). In: Motives (Seattle,WA, 1991), Proc. Sympos. Pure Math., 55, American Mathematical Society, Providence, RI, 1994, pp. 639–676.Google Scholar
[10] Ribet, K. A., Images of semistable Galois representations. Olga Taussky-Todd: in memoriam. Pacific J. Math. 1997, Special Issue, 277–297. http://dx.doi.org/10.2140/pjm.1997.181.277 Google Scholar
[11] Rubin, K., Modularity of mod 5 representations. In: Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 463–474.Google Scholar
[12] Serre, J.-P., Sur les représentations modulaires de degré 2 de Gal(Q/Q).Duke Math. J. 54(1987), no. 1, 179–230. http://dx.doi.org/10.1215/S0012-7094-87-05413-5 Google Scholar
[13] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan, 11, Kanô Memorial Lectures, 1, Princeton University Press, Princeton, NJ, 1994.Google Scholar
[14] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141(1995), no. 3, 553–572. http://dx.doi.org/10.2307/2118560 Google Scholar
[15] Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141(1995), no. 3, 443–551. http://dx.doi.org/10.2307/2118559 Google Scholar