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The generalized Fermat equation with exponents 2, 3, n

Titelangaben

Freitas, Nuno ; Naskręcki, Bartosz ; Stoll, Michael:
The generalized Fermat equation with exponents 2, 3, n.
In: Compositio Mathematica. Bd. 156 (2020) Heft 1 . - S. 77-113.
ISSN 1570-5846
DOI: https://doi.org/10.1112/s0010437x19007693

Angaben zu Projekten

Projekttitel:
Offizieller Projekttitel
Projekt-ID
Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory
SPP 1489

Abstract

We study the generalized Fermat equation x²+y³=z^p, to be solved in coprime integers, where p⩾7 is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve X(p). We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic p-torsion modules. Using these criteria we produce the minimal list of twists of X(p) that have to be considered, based on local information at 2 and 3; this list depends on p mod 24. We solve the equation completely when p=11, which previously was the smallest unresolved p. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on X₀(11) defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case p=13. The source code for the various computations is supplied as supplementary material with the online version of this article.

Weitere Angaben

Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
Keywords: generalized Fermat equation; modularity; symplectic method; rational points; Selmer group Chabauty
Fachklassifikationen: 2010 Mathematics Subject Classification: 11D41 (primary), 11F80, 11G05, 11G07, 11G30, 14G05, 14G25 (secondary)
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik II (Computeralgebra)
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik II (Computeralgebra) > Lehrstuhl Mathematik II (Computeralgebra) - Univ.-Prof. Dr. Michael Stoll
Fakultäten
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 14 Okt 2020 10:04
Letzte Änderung: 26 Sep 2023 10:51
URI: https://eref.uni-bayreuth.de/id/eprint/58386