If
S
0
,
S
1
,
S
2
S_{0}, S_{1}, S_{2}
are connected Riemann surfaces,
β
1
:
S
1
→
S
0
\beta _{1}:S_{1} \to S_{0}
and
β
2
:
S
2
→
S
0
\beta _{2}:S_{2} \to S_{0}
are surjective holomorphic maps, then the associated fiber product
S
1
×
(
β
1
,
β
2
)
S
2
S_{1} \times _{(\beta _{1},\beta _{2})} S_{2}
has the structure of a one-dimensional complex analytic space, endowed with a canonical map
β
:
S
1
×
(
β
1
,
β
2
)
S
2
→
S
0
\beta : S_{1} \times _{(\beta _{1},\beta _{2})} S_{2} \to S_{0}
, such that, for
j
=
1
,
2
j=1,2
,
β
j
∘
π
j
=
β
\beta _{j} \circ \pi _{j}=\beta
, where
π
j
:
S
1
×
(
β
1
,
β
2
)
S
2
→
S
j
\pi _{j}: S_{1} \times _{(\beta _{1},\beta _{2})} S_{2} \to S_{j}
is the natural coordinate projection. The connected components of the complement of its singular locus provide its irreducible components. A Fuchsian description of the irreducible components of
S
1
×
(
β
1
,
β
2
)
S
2
S_{1} \times _{(\beta _{1},\beta _{2})} S_{2}
is provided and, as a consequence, we obtain that if one of the maps
β
j
\beta _{j}
is a regular branched covering, then all its irreducible components are isomorphic. Also, if both
β
1
\beta _{1}
and
β
2
\beta _{2}
are of finite degree, then we observe that the number of these irreducible components is bounded above by the greatest common divisor of the two degrees, and that such an upper bound is sharp. We also provide sufficient conditions for the irreducibility of the connected components of the fiber product.
In the case that
S
0
=
C
^
S_{0}=\widehat {\mathbb C}
, and
S
1
S_{1}
and
S
2
S_{2}
are compact, we define the strong field of moduli of the pair
(
S
1
×
(
β
1
,
β
2
)
S
2
,
β
)
(S_{1} \times _{(\beta _{1},\beta _{2})} S_{2},\beta )
and observe that this field coincides with the minimal field containing the fields of moduli of both pairs
(
S
1
,
β
1
)
(S_{1},\beta _{1})
and
(
S
2
,
β
2
)
(S_{2},\beta _{2})
. Finally, in the case that all surfaces
S
1
S_{1}
,
S
2
S_{2}
and
S
0
S_{0}
are compact and the fiber product is a connected Riemann surface, we observe that the Jacobian variety
J
(
S
1
×
(
β
1
,
β
2
)
S
2
)
×
J
S
0
J(S_{1} \times _{(\beta _{1},\beta _{2})}S_{2}) \times JS_{0}
is isogenous to
J
S
1
×
J
S
2
×
P
JS_{1} \times JS_{2} \times P
, where
P
P
is a suitable abelian subvariety of
J
(
S
1
×
(
β
1
,
β
2
)
S
2
)
J (S_{1} \times _{(\beta _{1},\beta _{2})}S_{2})
.