Talk on Siegel theta series and modular forms

overview

• Siegel theta series
• Siegel modular forms
• Siegel-Weil formula

modular forms

• $\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}$
• 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 $\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})$

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

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

Eisenstein series

• for an integer $k\geq 2$, define the Eisenstein series by

$$ E_{2k}(\tau) : =\frac{1}{2}\sum_{ \substack{ (c,d)\in \mathbb{Z}^2\\ (c,d)=1 }} \frac{1}{(c\tau+d )^{2k}} $$

• Fourier expansion

\[E_{2k}(\tau):= 1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)=1-\frac {4k}{B_{2k}}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)\] where $\zeta$ denotes the Riemann zeta function, $B_k$ Bernoulli number and $\sigma_r(n)=\sum_{d|n}d^r$

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

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

the space of modular forms

thm

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

• dimension generating function

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

theta functions

notation

• $\Lambda\subset \mathbb{R}^n$ : integral lattice, i.e. a free abelian group with a positive definite symmetric bilinear form, i.e. $x\cdot y\in \mathbb{Z}$ for all $x,y\in \Lambda$
• we will assume that $\Lambda$ is even, i.e., $x\cdot x\in 2\mathbb{Z}$
• for a basis of $\Lambda$, fix $M$, $n\times n$ matrix whose each row is a basis element
• $A:=M^tM$, Gram matrix of $\Lambda$

definition

• old problem in number theory : find the number of representations of a given integer by the quadratic form associated to $\Lambda$
• for a given integer $N$, determine the size of the set $\{x\in\Lambda|x\cdot x=2N\}$ or $\{\zeta\in \mathbb{Z}^n|\zeta A \zeta^{t} =2N\}$
• denote it by $a(N)$
• theta function of $\Lambda$ is a holomorphic function on $\mathbb{H}$ given by

$$ \Theta_\Lambda(\tau)=\sum_{x\in\Lambda}q^{\frac{x\cdot x}{2}}=\sum_{N=0}^\infty a(N)q^{N}, $$ where $q=e^{2\pi i \tau}$

on theta functions of positive definite even unimodular lattices

8차원

• $\dim M_4=1$ and thus

$$\theta_{E_8}(\tau)=E_4(\tau)=1+240 q+2160 q^2+6720 q^3+17520 q^4+30240 q^5+\cdots$$

16차원

• $\dim M_8=1$, $E_8=E_4^2$ and

$$ \theta_{E_8\oplus E_8}(\tau)=\theta_{D_{16}^{+}}(\tau)=E_8(\tau)\\ E_8(\tau)=1+480 q+61920 q^2+1050240 q^3+7926240 q^4+\cdots $$

24차원

• 틀:수학노트의 세타함수
• modular form of weight 12
• $M_{12}=\mathbb{C}\langle E_4^3,E_6^2\rangle$
• let ${\rm gen}(L)$ be the set of all isomorphim classes of 24-dimensional positive definite even unimodular lattices
• to compute $\theta_{\Lambda}$, find $a,b$ such that $\theta_{\Lambda}=a E_4^3+ bE_6^2$
• we can easily determine $a,b$ once we know the number $r$ of roots in $\Lambda$ (the coefficient of $q$ in $\theta_{\Lambda}$) by solving

$$ \left\{ \begin{array}{c} a+b=1 \\ 720 a - 1008 b=r \end{array} \right. $$

• weighted average

$$\left(\sum_{\Lambda\in {\rm gen}(L)}\frac{\Theta_{\Lambda}(\tau)}{|{\rm Aut}(\Lambda)|}\right)\,\cdot\, \left(\sum_{\Lambda\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(\Lambda)|}\right)^{-1}=?$$

• we get

$$\left( \sum_{\Lambda\in {\rm gen}(L)}\frac{\Theta_{\Lambda}(\tau)}{|{\rm Aut}(\Lambda)|}\right)\,\cdot\, \left(\sum_{\Lambda\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(\Lambda)|}\right)^{-1}=E_{12}(\tau)$$ where $E_{12}$ is the Eisenstein series $$ E_{12}(\tau)=1+\frac{65520 q}{691}+\frac{134250480 q^2}{691}+\frac{11606736960 q^3}{691}+\frac{274945048560 q^4}{691}+\frac{3199218815520 q^5}{691}+\cdots $$

Siegel theta series

• 틀:수학노트
• for $g\in \mathbb{N}$ and $\Lambda$ of rank $n$, we will define the Siegel theta series $\Theta_\Lambda^{(g)}$ of degree (or genus) $g$ ($g$ comes from the genus of Riemann surfaces)
• $g=1$ case recovers $\Theta_\Lambda^{(1)}=\Theta_\Lambda$

def (half-integral matrix)

A symmetric matrix $N\in \operatorname{GL}(g,\mathbb{Q})$ is called half-integral if $2N$ has integral entries with even integers on the diagonal

representations of a quadratic form by another quadratic form

• we want to find the number of representations of a quadratic form by the quadratic form of $\Lambda$
• let $g\leq n$
• $\underline{x}$ : $g\times n$ matrix whose row is an element of $\Lambda$
• for each half-integral $g\times g$ matrix $\underline{N}=(N_{ij})$, let $a(\underline{N})$ be the number of elements in $\{\underline{x}=(x_i)\in\Lambda^{g}| x_i\cdot x_j=2N_{ij}\}$
• a given $\underline{x}$ can be written as $\underline{x}=\underline{\zeta}M$ for some $\underline{\zeta}$, a $g\times n$ integer matrix
• $a(\underline{N})$ is the number of elements in $\{\underline{\zeta}\in\mathbb{Z}^{g,n}|\underline{\zeta} A \underline{\zeta}^t =2\underline{N}\}$

definition

• Let $\tau=(\tau_{ij})$ be a symmetric $g\times g$ matrix
• for $\Lambda$, the theta series $\Theta_\Lambda^{(g)}$ of genus $g$ is defined by

$$ \begin{align} \Theta_\Lambda^{(g)}(\tau)&=\sum_{\underline{x}\in\Lambda^{g}}e^{\pi i\operatorname{Tr}(\underline{x}\cdot \underline{x} \tau)}\\ &=\sum_{\underline{\zeta}\in\mathbb{Z}^{g,n}}e^{\pi i\operatorname{Tr}(\underline{\zeta} A \underline{\zeta}^{t}\tau)}\\ &=\sum_{\underline{N}:\text{h.i.}} a(\underline{N})e^{2\pi i\operatorname{Tr}(\underline{N}\tau)} \end{align} \label{tg} $$

note on trace

• in the last equality, we used the following property of trace
• for two $n\times n$ matrices $A=(a_{ij})$ and $B=(b_{ij})$,

$$ \operatorname{tr}(AB)=\sum_{i,j=1}^{n}a_{ij}b_{ji} $$

• if $A$ and $B$ are symmetric,

$$ \operatorname{tr}(AB)=\sum_{i,j=1}^{n}a_{ij}b_{ij} $$

• the series \ref{tg} converges absolutely if $\tau$ is an element of

$$ \mathcal{H}_g:=\left\{\tau \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} $$

• it is a holomorphic function on $\mathcal{H}_g$

Siegel theta functions of even unimodular lattices

8차원

• $g=2$ case
• Fourier coefficient of $\Theta_{E_8}^{(2)}$
• $N = \Bigl( {a \atop b/2} \thinspace {b/2 \atop c} \Bigr) \in

\operatorname{Mat}_{2\times 2}({1 \over 2}\Z)$, positive semi-definite, half-integral matrix

• for $\tau=\left(

\begin{array}{cc} \tau _1 & z \\ z & \tau _2 \end{array} \right)$, $$ \operatorname{Tr}(N\tau)=a \tau _1+b z+c \tau _2 $$

• by setting $q_i=e^{2\pi i \tau_i}$, $\zeta=e^{2\pi i z}$, we get

$$\exp(2\pi i \operatorname{Tr}(N\tau))=q_1^a\zeta^bq_2^c$$

• let us compute $a(N)$ for $N=

\left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right), \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right), \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$.

• for the third one, we may use the following property of the $E_8$ root system $\Phi$

1. for a given $v\in \Phi$, there exist 126 elements in $\Phi$ orthogonal to $v$
2. 240*126=30240

• table

$$ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} N & \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 2 \end{array} \right) & \left( \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right) \\ \hline a(N) & 1 & 240 & 240 & 2160 & 2160 & 240 & 13440 & 30240 & 13440 & 240 \\ \hline \exp(2\pi i \operatorname{Tr}(N\tau)) & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2 \end{array} $$

16차원

• $E_8\oplus E_8$ and $D_{16}^{+}$ lattice
• for $g=1,2,3$, $\Theta_{E_8\oplus E_8}^{(g)}=\Theta_{D_{16}^{+}}^{(g)}$
• $\Theta^{(4)}_{E_8\oplus E_8}\neq \Theta^{(4)}_{D_{16}^{+}}$
• $\Theta^{(4)}_{E_8\oplus E_8}-\Theta^{(4)}_{D_{16}^{+}}$, Siegel cusp form of weight 8 called the Schottky form

24차원

• for 24 Niemeier lattices, the associated theta series are linearly dependent in degree $\leq$ 11 and linearly independent in degree 12 (Borcherds-Freitag-Weissauer, 1998)

thm

For a positive definite even unimodular lattice $\Lambda$, $\theta^{(g)}_{\Lambda}$ is a Siegel modular form of weight $\frac{n}{2}$ w.r.t. $\Gamma_g$

symplectic group

• symplectic group $\Gamma_g:=\operatorname{Sp}(2g,\Z)=\{M\in \operatorname{GL}(2g,\mathbb{Z})|M^T J_{g} M = J_{g}\}$

where $$ J_{g} =\begin{pmatrix}0 & I_g \\-I_g & 0 \\\end{pmatrix} $$

• $2g\times 2g$ matrix
• one can check that for

$$M=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g,$$ $$ \begin{align} A^tC=C^tA \\ B^tD=D^tB \\ A^tD-C^tB= I_g \end{align} $$

• the lattice $\mathbb{Z}^{2g}$ of rank $2g$ with basis $a_1,\cdots, a_g,b_1\cdots,b_g$ with the symplectic form

$$ \langle a_i,b_j \rangle = \begin{cases} 1, & \text{if }i=j\\ 0, & \text{if }i\neq j \\ \end{cases} $$

• then $\Gamma_g=\operatorname{Aut}(\mathbb{Z}^{2g},\langle,\rangle)$
• note that

$$ \begin{pmatrix} I_g & S \\ 0& I_g \\\end{pmatrix} \in \Gamma_g $$ for any symmetric integral matrix $S$

Siegel upper-half space

• $\mathcal{H}_g$

$$ \mathcal{H}_g=\left\{\tau \in \operatorname{Mat}_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} $$

• there is an action of $\Gamma_g$ on $\mathcal{H}_g$ by

$$ \tau\mapsto \gamma(\tau)=(A\tau +B)(C\tau + D)^{-1} $$

• we need to check that $C\tau + D$ Is invertible and $\Im{\gamma(\tau)}>0 $

Riemann bilinear relation

• 틀:수학노트
• $X$ : compact Riemann surface of genus $g$
• there exists a basis \(a_1, \dots, a_g,b_1,\cdots,b_g\) of \(H_1(X, \mathbb{Z}) \cong \mathbb{Z}^{2g}\) with the intersection pairing (canonical homology basis)

$$ \langle a_i,b_j \rangle = \begin{cases} 1, & \text{if }i=j\\ 0, & \text{if }i\neq j \\ \end{cases} $$

• there exists a basis of the space of holomorphic 1-form, $\omega_1,\cdots,\omega_{g}$ such that

$$ \int_{a_i}\omega_j=\delta_{ij} $$

• if we set $\tau_{i,j}=\int_{b_i}\omega_j$, then $\tau=(\tau_{i,j})_{1\leq i,j\leq g}$ satisfies the following properties

1. $\tau^{\mathrm{T}}=\tau$
2. $\textrm{Im}(\tau)$ is positive definite

• this is called the Riemann bilinear relation
• $\tau\in \mathcal{H}_g$ and and it is called a period matrix of $X$
• $\mathcal{A}_g=\mathcal{H}_g/\Gamma_g$ : moduli space of principally polarized abelian varieties

Siegel modular forms

• 틀:수학노트

definition

A holomorphic function $f:\mathcal{H}_g\to \mathbb{C}$ is a Siegel modular form of weight k and genus(or degree) $g$ if $$ f \left( (A\tau +B)(C\tau + D)^{-1}\right) = \det(C\tau +D)^{k} f(\tau),\, \forall \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g $$ and it must be holomorphic at the cusp if $g=1$

• denote the vector space of such functions as $M_k(\Gamma_g)$

Fourier expansion

• note that

$$ \begin{pmatrix} I_g & S \\ 0& I_g \\\end{pmatrix}\cdot \tau = \tau+S $$

• $f\in M_k(\Gamma_g)$ satisfies $f(\tau+S)=f(\tau)$ for any symmetric integral $S$
• we get the following expansion

$$ f(q_{11},\cdots, q_{gg})=\sum_{n_{11},\cdots, n_{ij},\cdots, n_{gg}\in \mathbb{Z}}a(n_{11},\cdots, n_{gg})q_{11}^{n_{11}}\cdots q_{gg}^{n_{gg}} \label{fou1} $$ where $q_{ij}=e^{2\pi i \tau_{ij}}$, $i\leq j$

• define a symmetric matrix $N=(N_{ij})_{1\leq i,j\leq g}$ as

$$ N_{ij}= \begin{cases} n_{ii}, & \text{if $i=j$}\\ n_{ij}/2, & \text{if $i\neq j$} \end{cases} $$

• $\operatorname{Tr}(N\tau)=\sum_{i=1}^{g}N_{ii}\tau_{ii}+2\sum_{1\leq i<j\leq g}N_{ij}\tau_{ij}$
• $\exp(2\pi i \operatorname{Tr}(N\tau))=q_{11}^{n_{11}}\cdots q_{gg}^{n_{gg}}$
• \ref{fou1} can be rewritten as

$$f(\tau)=\sum_{N}a(N)\exp\left(2\pi i \operatorname{Tr}(N\tau)\right)$$ where the summation is over $N=(N_{ij})\in \operatorname{Mat}_g(\frac{1}{2}\mathbb{Z})$ half-integral matrix

Koecher Principle

For a Siegel modular form $f\in M_k(\Gamma_g)$, if $N$ is not a positive semi-definite matrix, then $a(N)=0$. (this is why holomorphicity at the cusp is not necessary if $g>1$)

지겔 모듈라 형식의 예

• 격자의 지겔 세타 급수
• 틀:수학노트

$$ E_{k}^{(g)}(\tau) = \sum_{(C,D)} \frac{1}{\det(C\tau +D)^{k}} $$ where the summation is over all $$ \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_{g,0}\backslash \Gamma_{g} $$ and $$ \Gamma_{g,0}=\{\begin{pmatrix}A & B \\ 0 & D \\\end{pmatrix}\in \Gamma_{g}\} $$ (the summation extends over all classes of coprime symmetric pairs, i. e. over all inequivalent bottom rows of elements of $\Gamma_g$ with respect to left multiplications by unimodular integer matrices of degree $g$. In other words, the sum is over a full set of representatives for the cosets $\operatorname{GL}(g,\mathbb{Z})\backslash \Gamma_{g}$)

• Fourier coefficients of Siegel-Eisenstein series

Siegel-Weil formula

• Siegel-Weil formula
• 틀:수학노트

thm

For a positive definite even unimodular lattice $L$, $$\left( \sum_{M\in {\rm gen}(L)}\frac{\Theta_M^{(g)}(Z)}{|{\rm Aut}(M)|}\right)\,\cdot\, \left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1}= E^{(g)}_{k}(Z),$$

Moreover, the Fourier coefficients $a_{E}(N)$ of $E$ can be expressed as an infinite product of local densities $$ a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp} $$

mass formula

• for a half-integral $N$,

$$ a_{E}(N)=\left( \sum_{M\in {\rm gen}(L)}\frac{r_M(N)}{|{\rm Aut}(M)|}\right)\,\cdot\, \left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ where $\Theta_M^{(g)}(Z)=\sum_{N}r_M(N)\exp\left(2\pi i \operatorname{Tr}(N\tau)\right)$

• if $2N$ is a Gram matrix of $L$, then we obtain

$$ a_{E}(N)=\left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ as $$ r_M(N) = \begin{cases} |\operatorname{Aut}(L)|, & \text{if }L\sim M \\ 0, & \text{if }L\nsim M \\ \end{cases} $$

• then we can express

$$ a_{E}(N)=\left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} $$ in terms of local densities \ref{lp}, which gives the Smith-Minkowski-Siegel mass formula