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The generalized Fermat equation with exponents 2, 3, n

Title data

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

Project information

Project title:
Project's official titleProject's id
Algorithmic and Experimental Methods in Algebra, Geometry and Number TheorySPP 1489

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Keywords: generalized Fermat equation; modularity; symplectic method; rational points; Selmer group Chabauty
Subject classification: 2010 Mathematics Subject Classification: 11D41 (primary), 11F80, 11G05, 11G07, 11G30, 14G05, 14G25 (secondary)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 14 Oct 2020 10:04
Last Modified: 14 Oct 2020 10:04
URI: https://eref.uni-bayreuth.de/id/eprint/58386