= Solution
The effective group is torsion free and every <modular cusp> is regular in the preceding sense, so orders of a meromorphic weight-$k$ form are integers. At interior points use a local automorphy trivialization; at a <modular cusp> use the Fourier order of the appropriate slash transform in its <cusp width> coordinate. Let $C$ be the sum of all <modular cusp> points, each once, and set
$$
\boxed{D(f)=\sum_{P\in X(\Gamma_1(p))}\operatorname{ord}_P(f)P-C.}
$$
A <meromorphic function> $\varphi$ belongs to the <Riemann-Roch space> $\mathcal L(D(f))$ exactly when $\operatorname{div}\varphi+D(f)\ge0$. In the interior this requires $f\varphi$ to have no <pole>; at a <modular cusp> it requires order at least one. Conversely, the quotient of any weight-$k$ <cusp form> by $f$ is a meromorphic weight-zero function satisfying precisely those inequalities. This proves the <cusp-form divisor presentation>
$$
\boxed{S_k(\Gamma_1(p))=f\,\mathcal L(D(f)).}
$$
To compute the degree without imposing a valence formula as an extra assumption, use the meromorphic tensor differential $f^{12}(d\tau)^{6k}$. Its automorphy factors cancel. Its order at an interior point is $12\operatorname{ord}_P f$; at a <modular cusp> it is $12\operatorname{ord}_P f-6k$, since $d\tau$ is a nonzero constant times $dq_c/q_c$. A meromorphic section of the $6k$th <tensor power> of the <canonical bundle> has total divisor degree $6k(2g-2)$. The <regular-cusp valence formula on a torsion-free modular curve> is therefore
$$
\sum_P\operatorname{ord}_P f=\frac k2(2g-2+r_\infty)=\frac{kd}{12},\qquad
\deg D(f)=\frac{kd}{12}-r_\infty.
$$
For $k\ge3$, $\deg D(f)-(2g-2)=(k-2)d/12>0$. The <Riemann-Roch theorem> says $\ell(D)-\ell(K-D)=\deg D+1-g$, and a divisor of negative degree has no nonzero sections. Thus $\ell(K-D)=0$ and
$$
\dim S_k=\frac{kd}{12}-r_\infty+1-g=\frac{(k-1)d}{12}-\frac{r_\infty}{2}.
$$
Using $d=(p^2-1)/2$ and $r_\infty=p-1$ gives
$$
\boxed{\dim S_k(\Gamma_1(p))=\frac{p-1}{24}\bigl((p+1)(k-1)-12\bigr),\qquad k\ge3.}
$$
The canonical-degree and Riemann-Roch facts used here are general results for compact <Riemann surfaces>, as permitted.
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