Revisiting the equation $x^2+y^3=z^p$

TL;DR

This paper investigates rational points on certain modular curves related to the equation x^2 + y^3 = z^p, using local point criteria to analyze solutions for specific elliptic curves and primes.

Contribution

It applies a local point existence criterion to classify the rational points on modular curves associated with the Fermat-type equation, extending previous modular method results.

Findings

For certain elliptic curves with conductor 864, no local points exist at specific primes.

The paper determines when modular curves can be discarded based on local point criteria.

Some modular curves cannot be eliminated using local information alone.

Abstract

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskr\k{e}cki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information.