Let
p
≥
3
p \geq 3
be a prime. Let
E
/
Q
E/\mathbb {Q}
and
E
′
/
Q
E’/\mathbb {Q}
be elliptic curves with isomorphic
p
p
-torsion modules
E
[
p
]
E[p]
and
E
′
[
p
]
E’[p]
. Assume further that either (i) every
G
Q
G_\mathbb {Q}
-modules isomorphism
ϕ
:
E
[
p
]
→
E
′
[
p
]
\phi : E[p] \to E’[p]
admits a multiple
λ
⋅
ϕ
\lambda \cdot \phi
with
λ
∈
F
p
×
\lambda \in \mathbb {F}_p^\times
preserving the Weil pairing; or (ii) no
G
Q
G_\mathbb {Q}
-isomorphism
ϕ
:
E
[
p
]
→
E
′
[
p
]
\phi : E[p] \to E’[p]
preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii).
Our approach is to consider the problem locally at a prime
ℓ
≠
p
\ell \neq p
. Firstly, we determine the primes
ℓ
\ell
for which the local curves
E
/
Q
ℓ
E/\mathbb {Q}_\ell
and
E
′
/
Q
ℓ
E’/\mathbb {Q}_\ell
contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of
E
/
Q
ℓ
E/\mathbb {Q}_\ell
and
E
′
/
Q
ℓ
E’/\mathbb {Q}_\ell
, to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from
p
p
.
We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form
y
2
=
x
p
−
ℓ
y^2 = x^p - \ell
and
y
2
=
x
p
−
2
ℓ
y^2 = x^p - 2\ell
where
ℓ
\ell
is a prime; we also give incremental results on the Fermat equation
x
2
+
y
3
=
z
p
x^2 + y^3 = z^p
. As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the
p
p
-torsion of two non-isogenous elliptic curves defined over
Q
\mathbb {Q}
.