# 모듈라 형식(modular forms)

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## 개요

• 푸앵카레 상반평면에서 정의된 해석함수
• 모듈라 성질과 cusp에서의 푸리에전개를 가짐
• 정수론에서 많은 중요한 역할

### 기호

• $\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}$
• 모듈라 군(modular group) $\Gamma=SL(2, \mathbb Z) = \left \{ \left. \left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\right| a, b, c, d \in \mathbb Z,\ ad-bc = 1 \right \}$
• $\operatorname{PSL}(2,\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})/\{\pm I\}$ acts on $\mathbb{H}$ by

$\tau\mapsto\frac{a\tau+b}{c\tau+d}$ for $\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})$

• $SL(2, \mathbb Z)$ is generated by $S$ and $T$

$S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ $S: \tau\mapsto -1/\tau,T: \tau\mapsto \tau+1$

## 모듈라 형식

def

A holomorphic function $f:\mathbb{H}\to \mathbb{C}$ is a modular form of weight $k$ (w.r.t. $SL(2, \mathbb Z)$) if

1. $$f \left( \frac{ a\tau +b}{ c\tau + d} \right) = (c\tau +d)^{k} f(\tau)$$
2. $f$ is "holomorphic at the cusp", i.e. it has a Fourier expansion of the following form

$$f(\tau)=\sum_{n=0}^{\infty}c(n)e^{2\pi i n \tau}$$

## 예

$\Delta(\tau)=q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots$

### 아이젠슈타인 급수

$$G_{2k}(\tau) : =\sum_{(m,n)\in \mathbb{Z}^2\backslash{(0,0)}}\frac{1}{(m+n\tau )^{2k}}$$

• Eisenstein series : normalization of $G_{2k}$

$E_{2k}(\tau):=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)$ where $\zeta$ denotes the Riemann zeta function and $\sigma_r(n)=\sum_{d|n}d^r$

• this is a modular form of weight $2k$
• for example

$E_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}=1 + 240 q + 2160 q^2 + \cdots$ $E_6(\tau)=1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n}=1 - 504 q - 16632 q^2 - \cdots$

## 구조 정리

정리

$M_k$ be the space of modular forms of weight $k$ and $M:=\bigoplus_{k\in \mathbb{Z}_{\geq 0}} M_k$. We have $M=\mathbb{C}[E_4,E_6]$

• 차원 생성 함수

$$\sum_{k=0}^{\infty}\dim M_k x^k=\frac{1}{\left(1-x^4\right)\left(1-x^{6}\right)}=1+x^4+x^6+x^8+x^{10}+2 x^{12}+x^{14}+2 x^{16}+2 x^{18}+2 x^{20}+\cdots$$

• 가령 $$\{E_6^2, \Delta\}$$는 $M_{12}$의 기저가 된다. 여기서 $\Delta=E_4^3-E_6^2$

## 메모

$d(\frac{az+b}{cz+d})=\frac{(acz+ad-acz-bc)}{(cz+d)^2}dz=(cz+d)^{-2}dz$

• 마르틴 아티클러 (Martin Eichler)는 다음과 같은 말을 남김

There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms.

다섯개의 기본적인 산술적 연산이 있다 : 더하기, 빼기, 곱하기, 나누기, 그리고 ... 모듈라 형식.

## 수학용어번역

• modular - 대한수학회 수학용어집

## 관련논문

• Kevin Buzzard, Computing weight one modular forms over $\C$ and $\Fpbar$, arXiv:1205.5077 [math.NT], May 23 2012, http://arxiv.org/abs/1205.5077
• Kevin Buzzard, Alan Lauder, A computation of modular forms of weight one and small level, arXiv:1605.05346 [math.NT], May 17 2016, http://arxiv.org/abs/1605.05346
• Schulze-Pillot, Rainer, and Abdullah Yenirce. “Petersson Products of Bases of Spaces of Cusp Forms and Estimates for Fourier Coefficients.” arXiv:1602.01803 [math], February 4, 2016. http://arxiv.org/abs/1602.01803.
• Bellaiche, Joel, and Kannan Soundararajan. “The Number of Non-Zero Coefficients of Modular Forms (mod P).” arXiv:1508.02095 [math], August 9, 2015. http://arxiv.org/abs/1508.02095.
• Jorgenson, Jay, Lejla Smajlovic, and Holger Then. “Certain Aspects of Holomorphic Function Theory on Some Genus Zero Arithmetic Groups.” arXiv:1505.06042 [math], May 22, 2015. http://arxiv.org/abs/1505.06042.