We obtain additional Diophantine applications of the methods surrounding Darmon’s program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic curve over
Q
\mathbb {Q}
due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation
x
7
+
y
7
=
3
z
n
\begin{equation*} x^7 + y^7 = 3 z^n \end{equation*}
for all integers
n
≥
2
n \ge 2
. Moreover, we explain how the use of higher dimensional Frey abelian varieties allows a more efficient proof of this result due to additional structures that they afford, compared to using only Frey elliptic curves.
As a second application, we use some of these additional structures that Frey abelian varieties possess to show that a full resolution of the generalized Fermat equation
x
7
+
y
7
=
z
n
x^7 + y^7 = z^n
depends only on the Cartan case of Darmon’s big image conjecture. In the process, we solve the previous equation for solutions
(
a
,
b
,
c
)
(a,b,c)
such that
a
a
and
b
b
satisfy certain
2
2
- or
7
7
-adic conditions and all
n
≥
2
n \ge 2
.