The generalized Fermat equation with exponents 2, 3, n

TL;DR

This paper investigates the solutions of the generalized Fermat equation x^2 + y^3 = z^p for prime p ≥ 7, employing modularity, local criteria, and the Selmer group Chabauty method, and fully solves the case p=11.

Contribution

It introduces new local criteria for elliptic curve isomorphisms, reduces the problem to finite twists of X(p), and completely solves the equation for p=11, advancing understanding of this class of Diophantine equations.

Findings

Complete solution for p=11 case.

New criteria for elliptic curve isomorphisms at 2 and 3.

Reduction of problem to finite twists of X(p) based on local data.

Abstract

We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced 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 first 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 $p \bmod 24$. Recent results on mod $p$ representations with image in the normalizer of a split Cartan subgroup allow us to reduce the list further in some cases. Our second main result is the complete solution of the equation when $p = 11$, which previously was the smallest unresolved $p$. One relevant new ingredient is the use of the `Selmer group Chabauty' method introduced by the third author in recent work, 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 GRH, 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 $p = 13$.