##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Anthony Kling

#### U. Arizona

## Comparison of Integral Structures on the Space of Modular Forms of Full Level $N$

##### 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 results 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}}

### April 21, 2022

### 2:00 PM

Pre-talk at 1:20 PM

APM 6402 and Zoom;

See https://www.math.ucsd.edu/~nts

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