In this paper we show how to explicitly write down equations of hyperelliptic curves over
Q
\mathbb {Q}
such that for all odd primes
ℓ
\ell
the image of the mod
ℓ
\ell
Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the
ℓ
\ell
-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod
ℓ
\ell
Galois representations.
The main result of the paper is the following. Suppose
n
=
2
g
+
2
n=2g+2
is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than
n
n
(this hypothesis is expected to hold for all
g
≠
0
,
1
,
2
,
3
,
4
,
5
,
7
,
g\neq 0,1,2,3,4,5,7,
and
13
13
). Then there is an explicit
N
∈
Z
N\in \mathbb {Z}
and an explicit monic polynomial
f
0
(
x
)
∈
Z
[
x
]
f_0(x)\in \mathbb {Z}[x]
of degree
n
n
, such that the Jacobian
J
J
of every curve of the form
y
2
=
f
(
x
)
y^2=f(x)
has
Gal
(
Q
(
J
[
ℓ
]
)
/
Q
)
≅
GSp
2
g
(
F
ℓ
)
\operatorname {Gal}(\mathbb {Q}(J[\ell ])/\mathbb {Q})\cong \operatorname {GSp}_{2g}(\mathbb {F}_\ell )
for all odd primes
ℓ
\ell
and
Gal
(
Q
(
J
[
2
]
)
/
Q
)
≅
S
2
g
+
2
\operatorname {Gal}(\mathbb {Q}(J[2])/\mathbb {Q})\cong S_{2g+2}
, whenever
f
(
x
)
∈
Z
[
x
]
f(x)\in \mathbb {Z}[x]
is monic with
f
(
x
)
≡
f
0
(
x
)
mod
N
f(x)\equiv f_0(x) \bmod {N}
and with no roots of multiplicity greater than
2
2
in
F
¯
p
\overline {\mathbb {F}}_p
for any
p
∤
N
p\nmid N
.