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\begin{document}

\title[The equation $x^3+y^3=z^n$]{The generalized Fermat equation $\boldsymbol{x^3+y^3=z^n}$}
\author{Manvir Jaswal}
\thanks{How AI was used in this work is described under Declarations, p.~\pageref{sec:declarations}.}
\address{Toronto, Canada}
\email{manvirjaswal@icloud.com}
\subjclass[2020]{Primary 11D41; Secondary 11F80, 11R29, 11R42, 14D05, 11G25}
\keywords{Generalized Fermat equation, Frey representations, hypergeometric motives, compatible systems of Galois representations, ordinary representations, local generation, discriminant bounds}
\hypersetup{pdftitle={The generalized Fermat equation x\unichar{"00B3} + y\unichar{"00B3} = z\unichar{"207F}},pdfauthor={Manvir Jaswal}}

\begin{abstract}
We prove that the equations $x^3+y^3=z^n$ and $x^3+y^3=3z^n$ have no solution in coprime nonzero integers for any $n\ge3$.  For the first equation the open cases were prime exponents $p\equiv1\pmod 3$ above $10^9$ outside a set of congruence classes treated by Chen and Siksek; in these cases the solution $1^3+2^3=3^2$ blocks the modular method.  We replace Kraus's Frey curve by a hypergeometric motive of rank three over $K=\Q(\sqrt{-3})$, whose parameter at a solution is $3$-adically close enough to a point of maximally unipotent monodromy for inertia at $\sqrt{-3}$ to act unipotently; the parameter of $1^3+2^3=3^2$ is not.  By a theorem of Calegari, Emerton and Gee, the mod~$p$ representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification, and we show that the member at a prime above $3$ is irreducible and ordinary at $\sqrt{-3}$.  Ramification and unconditional discriminant bounds show that its reductions have trivial semisimplification, and as the Galois group of the maximal pro-$3$ extension of $K$ unramified outside $\sqrt{-3}$ is generated by one inertia group, the member stabilizes its ordinary line, a contradiction.  For $x^3+y^3=3z^n$ the same argument also works at the primes $p\equiv2\pmod3$, which are inert in $K$, and Frey curves and descents treat the remaining exponents.
\end{abstract}

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\input{sections/01-introduction}
\input{sections/02-conventions}
\part*{\texorpdfstring{\boldmath}{}Part I: a Frey motive of rank three over \texorpdfstring{$\Q(\sqrt{-3})$}{Q(\unichar{"221A}\unichar{"2212}3)}}
\input{sections/03-motive}
\input{sections/04-inertia3}
\input{sections/05-frey}
\input{sections/06-irreducibility}
\part*{\texorpdfstring{\boldmath}{}Part II: a compatible system and its members above $5$}
\input{sections/07-lift}
\input{sections/08-bounds}
\input{sections/09-reductions}
\part*{\texorpdfstring{\boldmath}{}Part III: the member at $3$, and the proof of Theorem~A}
\input{sections/10-three}
\part*{\texorpdfstring{\boldmath}{}Part IV: the equation \texorpdfstring{$x^3+y^3=3z^n$}{x\unichar{"00B3}+y\unichar{"00B3}=3z\unichar{"207F}}}
\input{sections/11-3zp}
\part*{Complements}
\input{sections/12-remarks}

\appendix
\crefalias{section}{appendix}\crefalias{subsection}{appendix}
\input{sections/13-computations}

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\end{document}