  Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric
over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite
set of points. We suppose that $g^{TM}$ is equal to the Poincar\'e metric over
a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the
twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm
$\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a
holomorphic Hermitian vector bundle over $\overline{M}$.
  In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes
\|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes
\omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that
$T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic
torsion of non-cusped surfaces. Then we prove the anomaly formula for the
associated Quillen norm. The results of this paper will be used in the sequel
to study the regularity of the Quillen norm and its asymptotics in a
degenerating family of Riemann surfaces with cusps and to prove the curvature
theorem. We also prove that our definition of the analytic torsion for
hyperbolic surfaces is compatible with the one obtained through Selberg trace
formula by Takhtajan-Zograf.
