Talk on Siegel theta series and modular forms

overview

Siegel theta series
Siegel modular forms
Siegel-Weil formula

modular forms

<math>\mathbb{H}=\{\tau\in \mathbb{C}|\Im \tau>0\}</math>
modular group <math>\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 \}</math>
<math>\operatorname{PSL}(2,\mathbb{Z})=\operatorname{SL}(2,\mathbb{Z})/\{\pm I\}</math> acts on <math>\mathbb{H}</math> by

<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math>

for <math>\left ( \begin{array}{cc}a & b \\ c & d \end{array} \right )\in \operatorname{SL}(2,\mathbb{Z})</math>

def

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

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

<math>

f(\tau)=\sum_{n=0}^{\infty}a(n)e^{2\pi i n \tau} </math>

Eisenstein series

for an integer <math>k\geq 2</math>, define the Eisenstein series by

<math>

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

Fourier expansion

<math>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)</math>

where <math>\zeta</math> denotes the Riemann zeta function, <math>B_k</math> Bernoulli number and <math>\sigma_r(n)=\sum_{d|n}d^r</math>

this is a modular form of weight <math>2k</math>
for example

<math>E_4(\tau)= 1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}=1 + 240 q + 2160 q^2 + \cdots </math>

<math>E_6(\tau)=1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n}=1 - 504 q - 16632 q^2 - \cdots </math>

the space of modular forms

thm

Let <math>M_k</math> be the space of modular forms of weight <math>k</math> and <math>M:=\bigoplus_{k\in \mathbb{Z}_{\geq 0}} M_k</math>. We have

<math>M=\mathbb{C}[E_4,E_6]</math>

dimension generating function

<math>

\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 </math>

theta functions

notation

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

definition

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

<math>

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

on theta functions of positive definite even unimodular lattices

8차원

<math>\dim M_4=1</math> and thus

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

16차원

<math>\dim M_8=1</math>, <math>E_8=E_4^2</math> and

<math>

\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 </math>

24차원

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

<math> \left\{ \begin{array}{c} a+b=1 \\ 720 a - 1008 b=r \end{array} \right. </math>

weighted average

<math>\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}=?</math>

we get

<math>\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)</math> where <math>E_{12}</math> is the Eisenstein series

<math>

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 </math>

Siegel theta series

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

def (half-integral matrix)

A symmetric matrix <math>N\in \operatorname{GL}(g,\mathbb{Q})</math> is called half-integral if <math>2N</math> 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 <math>\Lambda</math>
let <math>g\leq n</math>
<math>\underline{x}</math> : <math>g\times n</math> matrix whose row is an element of <math>\Lambda</math>
for each half-integral <math>g\times g</math> matrix <math>\underline{N}=(N_{ij})</math>, let <math>a(\underline{N})</math> be the number of elements in <math>\{\underline{x}=(x_i)\in\Lambda^{g}| x_i\cdot x_j=2N_{ij}\}</math>
a given <math>\underline{x}</math> can be written as <math>\underline{x}=\underline{\zeta}M</math> for some <math>\underline{\zeta}</math>, a <math>g\times n</math> integer matrix
<math>a(\underline{N})</math> is the number of elements in <math>\{\underline{\zeta}\in\mathbb{Z}^{g,n}|\underline{\zeta} A \underline{\zeta}^t =2\underline{N}\}</math>

definition

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

<math>

\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} </math>

note on trace

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

<math>

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

if <math>A</math> and <math>B</math> are symmetric,

<math>

\operatorname{tr}(AB)=\sum_{i,j=1}^{n}a_{ij}b_{ij} </math>

the series \ref{tg} converges absolutely if <math>\tau</math> is an element of

<math>

\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\} </math>

it is a holomorphic function on <math>\mathcal{H}_g</math>

Siegel theta functions of even unimodular lattices

8차원

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

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

for <math>\tau=\left(

\begin{array}{cc} \tau _1 & z \\ z & \tau _2 \end{array} \right)</math>,

<math>

\operatorname{Tr}(N\tau)=a \tau _1+b z+c \tau _2 </math>

by setting <math>q_i=e^{2\pi i \tau_i}</math>, <math>\zeta=e^{2\pi i z}</math>, we get

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

let us compute <math>a(N)</math> for <math>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)</math>.

for the third one, we may use the following property of the <math>E_8</math> root system <math>\Phi</math>

for a given <math>v\in \Phi</math>, there exist 126 elements in <math>\Phi</math> orthogonal to <math>v</math>
240*126=30240

table

<math>

\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} </math>

16차원

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

24차원

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

thm

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

symplectic group

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

where

<math>

J_{g} =\begin{pmatrix}0 & I_g \\-I_g & 0 \\\end{pmatrix} </math>

<math>2g\times 2g</math> matrix
one can check that for

<math>M=\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_g,</math>

<math>

\begin{align} A^tC=C^tA \\ B^tD=D^tB \\ A^tD-C^tB= I_g \end{align} </math>

the lattice <math>\mathbb{Z}^{2g}</math> of rank <math>2g</math> with basis <math>a_1,\cdots, a_g,b_1\cdots,b_g</math> with the symplectic form

<math>

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

then <math>\Gamma_g=\operatorname{Aut}(\mathbb{Z}^{2g},\langle,\rangle)</math>
note that

<math>

\begin{pmatrix} I_g & S \\ 0& I_g \\\end{pmatrix} \in \Gamma_g </math> for any symmetric integral matrix <math>S</math>

Siegel upper-half space

<math>\mathcal{H}_g</math>

<math>

\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\} </math>

there is an action of <math>\Gamma_g</math> on <math>\mathcal{H}_g</math> by

<math>

\tau\mapsto \gamma(\tau)=(A\tau +B)(C\tau + D)^{-1} </math>

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

Riemann bilinear relation

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

<math>

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

there exists a basis of the space of holomorphic 1-form, <math>\omega_1,\cdots,\omega_{g}</math> such that

<math>

\int_{a_i}\omega_j=\delta_{ij} </math>

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

<math>\tau^{\mathrm{T}}=\tau</math>
<math>\textrm{Im}(\tau)</math> is positive definite

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

Siegel modular forms

틀:수학노트

definition

A holomorphic function <math>f:\mathcal{H}_g\to \mathbb{C}</math> is a Siegel modular form of weight k and genus(or degree) <math>g</math> if

<math>

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 </math> and it must be holomorphic at the cusp if <math>g=1</math>

denote the vector space of such functions as <math>M_k(\Gamma_g)</math>

Fourier expansion

note that

<math>

\begin{pmatrix} I_g & S \\ 0& I_g \\\end{pmatrix}\cdot \tau = \tau+S </math>

<math>f\in M_k(\Gamma_g)</math> satisfies <math>f(\tau+S)=f(\tau)</math> for any symmetric integral <math>S</math>
we get the following expansion

<math>

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} </math> where <math>q_{ij}=e^{2\pi i \tau_{ij}}</math>, <math>i\leq j</math>

define a symmetric matrix <math>N=(N_{ij})_{1\leq i,j\leq g}</math> as

<math>

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

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

<math>f(\tau)=\sum_{N}a(N)\exp\left(2\pi i \operatorname{Tr}(N\tau)\right)</math>

where the summation is over <math>N=(N_{ij})\in \operatorname{Mat}_g(\frac{1}{2}\mathbb{Z})</math> half-integral matrix

Koecher Principle

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

지겔 모듈라 형식의 예

<math>

E_{k}^{(g)}(\tau) = \sum_{(C,D)} \frac{1}{\det(C\tau +D)^{k}} </math> where the summation is over all

<math>

\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in \Gamma_{g,0}\backslash \Gamma_{g} </math> and

<math>

\Gamma_{g,0}=\{\begin{pmatrix}A & B \\ 0 & D \\\end{pmatrix}\in \Gamma_{g}\} </math> (the summation extends over all classes of coprime symmetric pairs, i. e. over all inequivalent bottom rows of elements of <math>\Gamma_g</math> with respect to left multiplications by unimodular integer matrices of degree <math>g</math>. In other words, the sum is over a full set of representatives for the cosets <math>\operatorname{GL}(g,\mathbb{Z})\backslash \Gamma_{g}</math>)

Fourier coefficients of Siegel-Eisenstein series

Siegel-Weil formula

thm

For a positive definite even unimodular lattice <math>L</math>,

<math>\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),</math>

Moreover, the Fourier coefficients <math>a_{E}(N)</math> of <math>E</math> can be expressed as an infinite product of local densities

<math>

a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp} </math>

mass formula

for a half-integral <math>N</math>,

<math>

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} </math> where <math>\Theta_M^{(g)}(Z)=\sum_{N}r_M(N)\exp\left(2\pi i \operatorname{Tr}(N\tau)\right)</math>

if <math>2N</math> is a Gram matrix of <math>L</math>, then we obtain

<math>

a_{E}(N)=\left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1} </math> as

<math>

r_M(N) = \begin{cases} |\operatorname{Aut}(L)|, & \text{if }L\sim M \\ 0, & \text{if }L\nsim M \\ \end{cases} </math>

then we can express

<math>

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

Talk on Siegel theta series and modular forms

목차

overview

modular forms

Eisenstein series

the space of modular forms

theta functions

notation

definition

on theta functions of positive definite even unimodular lattices

8차원

16차원

24차원

Siegel theta series

representations of a quadratic form by another quadratic form

definition

note on trace

Siegel theta functions of even unimodular lattices

8차원

16차원

24차원

symplectic group

Siegel upper-half space

Riemann bilinear relation

Siegel modular forms

Fourier expansion

지겔 모듈라 형식의 예

Siegel-Weil formula

mass formula

둘러보기 메뉴

Talk on Siegel theta series and modular forms

overview

modular forms

Eisenstein series

the space of modular forms

theta functions

notation

definition

on theta functions of positive definite even unimodular lattices

8차원

16차원

24차원

Siegel theta series

representations of a quadratic form by another quadratic form

definition

note on trace

Siegel theta functions of even unimodular lattices

8차원

16차원

24차원

symplectic group

Siegel upper-half space

Riemann bilinear relation

Siegel modular forms

Fourier expansion

지겔 모듈라 형식의 예

Siegel-Weil formula

mass formula

둘러보기 메뉴

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