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The generalized Fermat equation with exponents 2, 3, $n$

Part of: Arithmetic algebraic geometry Diophantine equations Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 November 2019

Nuno Freitas ,
Bartosz Naskręcki  and
Michael Stoll
Nuno Freitas
Affiliation:
Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain email nunobfreitas@gmail.com
Bartosz Naskręcki
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland email nasqret@gmail.com
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany email Michael.Stoll@uni-bayreuth.de
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Abstract

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ 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 $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. 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 $X_{0}(11)$ 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.

Keywords

generalized Fermat equationmodularitysymplectic methodrational pointsSelmer group Chabauty

MSC classification

Primary: 11D41: Higher degree equations; Fermat's equation
Secondary: 11F80: Galois representations 11G05: Elliptic curves over global fields 11G07: Elliptic curves over local fields 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 14G05: Rational points 14G25: Global ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 77 - 113
Copyright
© The Authors 2019 

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Footnotes

The work reported on in this paper was supported by the Deutsche Forschungsgemeinschaft, grant Sto 299/11-1, in the framework of the Priority Programme SPP 1489. The first author was also partly supported by the grant Proyecto RSME-FBBVA 2015 José Luis Rubio de Francia.

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