Comparison of integral structures on the space of modular forms of full level N

Let $N\geq3$ and $r\geq1$ be integers and $p\geq2$ be a prime such that $p\nmid N$. One can consider two different integral structures on the space of modular forms over $\mathbb{Q}$, one coming from arithmetic via $q$-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level $\Gamma(Np^{r})$ over $\mathbb{Q}_{p}(\zeta_{Np^{r}})$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^{r}$ whenever $p^{r}>3$, allowing us to compute a lower bound which agrees with the upper bound. Hence we are able to compute the exponent precisely.