Cubic forms, powers of primes and classification of elliptic curves
Volume 166 / 2021
Colloquium Mathematicum 166 (2021), 137-150
MSC: Primary 11D41.
DOI: 10.4064/cm8132-8-2020
Published online: 11 March 2021
Abstract
We prove that for almost all primes $p$ the equation $(x+y)(x^2+Bxy+y^2)=p^\alpha z^n$ with $B\in \{0,1,4,6\}$ has no solution in pairwise coprime nonzero integers $x$, $y$, $z\neq \pm 1$, integer $\alpha \gt 0$ and prime $n\geq p^{16p}$. Proving this we classify primes $p$ for which there exists an elliptic curve over $\mathbb {Q}$ with conductor $48p$, $192p$ or $384p$ and with nontrivial rational $2$-torsion.
