math.NT — Number Theory

The generalized Fermat equation $x^3+y^3=z^n$

Contributed by Manvir Jaswal

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 $K=\mathbb{Q}(\sqrt{-3})$, 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.