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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 ofmodular forms over $\mathbb{Q}$, one coming from arithmetic via $q$-expansions,the other coming from geometry via integral models of modular curves. Bothstructures are stable under the Hecke operators; furthermore, their quotient isfinite torsion. Our goal is to investigate the exponent of the annihilator ofthe quotient. We will apply methods due to Brian Conrad to the situation ofmodular forms of even weight and level $\Gamma(Np^{r})$ over$\mathbb{Q}_{p}(\zeta_{Np^{r}})$ to obtain an upper bound for the exponent. Wealso 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 theupper bound. Hence we are able to compute the exponent precisely.

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