For
K
K
a number field and
p
{\mathbf {p}}
a finite prime of
K
K
, we define a
p
{\mathbf {p}}
-adic hyperbolic plane and study its geometry under the action of
G
L
2
(
K
p
)
G{L_2}({K_{\mathbf {p}}})
. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator
T
p
{T_{\mathbf {p}}}
. Letting
S
S
be a finite set of primes of a totally real
K
K
(including all the infinite ones), a modular group
Γ
(
S
)
\Gamma (S)
is defined. This group acts discontinuously on a product of classical and
p
{\mathbf {p}}
-adic hyperbolic planes.
S
S
-modular forms and their associated Dirichlet series are studied.