We shall explore the geometry of the Modular curve $X_0(p^2)$ and
it's regular minimal model over the ring of integers, which is an arithmetic surface.
After a base change we shall show that the regular minimal model is semi-simple.
Arakelov has introduced an intersection pairing for divisors on arithmetic surfaces.
We shall derive an expression for the Arakelov self-intersection of the relative
dualising sheaf on the regular minimal model of $X_0(p^2)$. As a consequence,
we shall give some number theoretic applications for this computation. This is
a joint work with Debargha Banerjee and Diganta Borah
