Title: The generalized Fermat equation with exponents 2, 3, n,
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 $p \bmod 24$. Using
recent results on mod $p$ representations with image in the normalizer of a
split Cartan subgroup, the list can be further reduced 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 a recent preprint, 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 $p = 13$.