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

TL;DR

The paper determines the exponent of the annihilator of the finite torsion quotient between two integral structures on spaces of modular forms of full level after restricting to primes dividing the level. It combines Conrad’s geometric intersection theory on arithmetic surfaces with explicit computations of the intersection matrix for the modular curve $\mathfrak{X}(Np^r)$ and the degree of the modular sheaf to obtain a sharp upper bound $e\le 2k p^{r-1}(pr-r+1)$; then it constructs explicit Klein-form modular forms to obtain a matching lower bound. For $p^r>3$ these bounds coincide, yielding an exact exponent, with two small-level exceptions noted. The approach provides a precise link between arithmetic q-expansions and geometric integral models, enhancing our understanding of integral structures and Hecke-stability in spaces of modular forms at full level $Np^r$.

Abstract

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 $Γ(Np^{r})$ over $\mathbb{Q}_{p}(ζ_{Np^{r}})$ 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.

Theorems & Definitions (179)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof
• Lemma 2.5