Talk on Siegel theta series and modular forms

수학노트
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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

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

\[ E_{k}^{(g)}(\tau) = \sum_{(C,D)} \frac{1}{\det(C\tau +D)^{k}} \] 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 \(\operatorname{GL}(g,\mathbb{Z})\backslash \Gamma_{g}\))

Siegel-Weil formula

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

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