"Talk on Gross-Keating invariants"의 두 판 사이의 차이

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−
==Siegel-Weil formula==
+
==overview==
 +
* defn of Gross-Keating inv. of a quad. form over Zp
 +
* bin. quad. forms and class number relations
 +
* representation of integers by quad. forms
 +
* (skip if no time) computer program that computes Gross-Keating inv. of a quad. form over Zp
 +
<math>
 +
\newcommand{\Z}{\mathbb Z}
 +
\newcommand{\Zn}{\Z_{\geq 0}^n}
 +
\newcommand{\Zp}{\mathbb {Z}_p}
 +
\newcommand{\matn}{\calh_n(\Zp)^{\rm nd}}
 +
\def\Zmat#1{\calh_{#1}(\Z)^{\rm nd}}
 +
\def\mat#1#2{\calh_{#1}(\Z_{#2})^{\rm nd}}
 +
\newcommand{\ord}{\mathrm{ord}}
 +
\newcommand{\calh}{\mathcal H}
 +
\newcommand{\frko}{\mathfrak o}
 +
\newcommand{\GL}{{\mathrm{GL}}}
 +
\newcommand{\GK}{\mathrm{GK}}
 +
\newcommand{\vep}{\varepsilon}
 +
\newcommand{\intmult}{(T_{m_1} \cdot T_{m_2}\cdot T_{m_3})_{S}}
 +
\newcommand{\Qp}{\mathbb {Q}_p}
 +
\newcommand{\diag}{\mathrm{diag}}
 +
\def\sym#1{{\rm Sym}_n(#1)}
 +
\newcommand\supparen[1]{^{(#1)}}
 +
\newcommand\Egk{E_k\supparen g}
 +
\newcommand\GLnZ{\GL n\Z}
 +
\newcommand\Xgsemi{\siX g^{\rm semi}}
 +
\newcommand\fc[2]{a(#1;#2)}
 +
\newcommand\e{\operatorname{e}}
 +
\newcommand\ip[2]{\langle #1,#2\rangle}
 +
\newcommand\siX[1]{{\mathcal X}_{#1}}
 +
\newcommand\Xn{\siX n}
 +
\newcommand\Xm{\siX m}
 +
\newcommand\Xg{\siX g}
 +
\newcommand\Xnsemi{\siX n^{\rm semi}}
 +
\newcommand\Xgsemi{\siX g^{\rm semi}}
 +
\newcommand\Xtwo{\siX 2^{\rm semi}}
 +
\newcommand\hh[1]{\mathbb{H}_{#1}}
 +
</math>
 +
 
 +
==Gross-Keating inv.==
 +
* [[Gross-Keating invariants of a quadratic form]]
 +
* <math>p\in \Z_{> 0}</math> : prime
 +
* <math>\Qp</math> : <math>p</math>-adic completion of <math>\Q</math>, and <math>\Zp</math> : ring of int.
 +
* For <math>a\in \Qp^\times</math>, <math>\ord(a)=n</math> if <math>a\in p^n \Zp^\times</math>, <math>\ord(0)=\infty</math>
 +
* symm. <math>n\times n</math> mat. <math>B=(b_{ij}),\, b_{ij}\in \Qp</math> is half-integral if <math>b_{ii}\in \Zp</math> and <math>2b_{ij}\in \Zp</math>
 +
* <math>\matn</math> : set of <math>n\times n</math> non-deg. half-int. mat.
 +
 
 +
;def
 +
<math>B=(b_{ij})\in\matn</math>
 +
 
 +
<math>S(B)</math> : set of all non-decreasing seq. <math>(a_1, \dots, a_n)\in\Zn</math> s.t.
 +
\begin{align*}
 +
&\ord(b_{ii})\geq a_i \qquad\qquad\qquad\quad (1\leq i\leq n), \\
 +
&\ord(2 b_{ij})\geq (a_i+a_j)/2  \qquad\; (1\leq i\leq j\leq n),
 +
\end{align*}
 +
 
 +
<math>S(\{B\}):=\bigcup_{U\in\GL_n(\Zp)} S(U^{t}BU)</math>
 +
 
 +
GK inv. <math>\GK(B)=(a_1, \dots, a_n)\in\Zn</math> of <math>B</math> is
 +
\begin{align*}
 +
a_1&=\max_{(y_1, \dots)\in S(\{B\})} \,y_1, \\
 +
a_2&=\max_{(a_1, y_2, \dots)\in S(\{B\})}\, y_2, \\
 +
&\dots \\
 +
a_n&=\max_{(a_1, a_2, \dots, a_{n-1}, y_n)\in S(\{B\})}\, y_n.
 +
\end{align*}
 +
 
 +
* By definition <math>GK(B)</math> depends only on <math>\Zp</math>-class of <math>B</math> under <math>B\sim B'</math> if <math>B' = U^{t}BU</math> for some <math>U\in\GL_n(\Zp)</math>
 +
* hard to compute from definition
 +
 
 +
;remarks
 +
* 1993 : Gross-Keating : introduced <math>GK(B)</math> for 3x3 <math>B</math> in study of arithmetic intersection number related to three modular poly.
 +
* 2015 : Ikeda-Katsurada : defined <math>GK(B)</math> for <math>B</math> <math>n\times n</math> half-int. over a finite ext'n of <math>\Qp</math>
 +
* 2016 : Ikeda-Katsurada : Siegel series of <math>B</math> (local factor of Fourier coef of Siegel-Eisenstein series) is determined by <math>GK(B)</math>
 +
* 2017 : Cho-Ikeda-Katsurada-Yamauchi : computer-friendly (not human-friendly) inductive formulas for <math>GK(B)</math>
 +
* I recently wrote computer program using Mathematica; arXiv:1809.10323
 +
 
 +
==bin. quad. forms and class number relations==
 +
* <math>Q=Ax^2+Bxy+Cy^2</math> : pos. def. bin. quad. form over <math>\Z</math>, write <math>Q=[A,B,C]</math>
 +
* disc. of <math>Q</math> :  <math>B^2-4AC<0</math>
 +
* for int. <math>d>0</math>,
 +
** <math>\mathcal{Q}_d=\{Q:B^2-4AC=-d\}</math>
 +
** <math>\mathcal{Q}_{d}^{pr}=\{Q\in \mathcal{Q}_d:\text{primitive}\}</math>. <math>Q</math> is prim. if <math>\rm{GCD}(A,B,C)=1</math>
 +
* <math>\Gamma=PSL_2(\mathbb{Z})</math> acts on <math>\mathcal{Q}_d</math> (and <math>\mathcal{Q}_{d}^{pr}</math>) : <math>Q\mapsto Q'</math> by <math>Q'(x,y)=Q(ax+by,cx+dy)</math>
 +
:<math>
 +
\left(
 +
\begin{array}{cc}
 +
A & \frac{B}{2} \\
 +
\frac{B}{2} & C \\
 +
\end{array}
 +
\right)
 +
\mapsto
 +
\left(
 +
\begin{array}{cc}
 +
a & b \\
 +
c & d \\
 +
\end{array}
 +
\right)^t\left(
 +
\begin{array}{cc}
 +
A & \frac{B}{2} \\
 +
\frac{B}{2} & C \\
 +
\end{array}
 +
\right)\left(
 +
\begin{array}{cc}
 +
a & b \\
 +
c & d \\
 +
\end{array}
 +
\right)
 +
</math>
 +
* for each <math>Q</math>, <math>w_{Q}</math> : size of stabilizers
 +
** <math>w_Q=2</math> if <math>Q\sim [a,0,a]</math>
 +
** <math>w_Q=3</math> if <math>Q\sim [a,a,a]</math>
 +
** <math>w_Q=1</math> otherwise
 +
;def (class number and Hurwitz-Kronecker class number)
 +
For int. <math>d>0</math>,
 +
:<math>h_{d}^{pr}:=\#(\mathcal{Q}_d^{pr}/\Gamma),\quad h_d:=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}</math>
 +
 
 +
;example
 +
* <math>\mathcal{Q}_{12}^{pr}/\Gamma = \{[1,0,3]\}</math>, <math>h_{12}^{pr} = 1</math>
 +
* <math>\mathcal{Q}_{12}/\Gamma = \{[1,0,3],[2,2,2]\}</math>, <math>h_{12} = 4/3</math>
 +
* when <math>d=23</math>, both are : <math>\{[1,1,6], [2,-1,3], [2,1,3]\}</math>, <math>h_{23}=h_{23}^{pr} = 3</math>
 +
 
 +
 
 +
===class poly===
 +
;def (j-inv.)
 +
:<math>
 +
j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+\cdots,\, q=e^{2\pi i\tau},\tau\in \mathbb{H}
 +
</math>
 +
where
 +
:<math> E_ 4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n,\quad \sigma_3(n)=\sum_{d|n}d^3</math>
 +
:<math>\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}</math>
 +
 
 +
 
 
;thm
 
;thm
−
For a positive definite even unimodular lattice $L$,
+
<math>Q=[A,B,C]</math> : prim of disc <math>-d</math>, and <math>\tau_Q = \frac{-B+\sqrt{B^2-4AC}}{2A}\in \mathbb{H}</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}=
+
Then <math>j(\tau_Q)</math> is an alg. int. with minimal poly.
−
E^{(g)}_{k}(Z),$$
+
:<math>
 +
H_d(x) : = \prod_{Q\in \mathcal{Q}_{d}^{\rm{pr}}/\Gamma}(x-j(\tau_Q))\in \Z[x]
 +
</math>
 +
In particular, <math>h_{d}^{\rm{pr}}=1</math>, then <math>j(\tau_Q)\in \mathbb{Z}</math>.
 +
 
 +
;example
 +
<math>h_{23}^{pr}=3, \qquad H_{23}(x) = x^3+3491750 x^2-5151296875 x+12771880859375</math>
 +
 
 +
===modular poly===
 +
* <math>m>0</math> : int
 +
* <math>\exists</math> <math>\phi_m(x,y)\in{\mathbb{Z}}[x,y]</math> such that
 +
:<math>\prod_{ad=m,a,d>0,0\leq b \leq d-1}(x-j(\frac{a\tau+b}{d}))=\phi_m(x,j(\tau))</math>
 +
* <math>\phi_m(j(m\tau),j(\tau))=0</math>
 +
* as a poly. in <math>x</math>, <math>\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d</math>
 +
 
 +
;examples
 +
* <math>m=1</math>,  <math>\phi_1(x,y)=x-y</math>
 +
* <math>m=2</math>
 +
:<math>
 +
\phi_2(x,y)=x^3+y^3-x^2 y^2+1488 (x^2 y + x y^2)-162000 (x^2+y^2) +40773375 x y+8748000000 (x + y)-157464000000000
 +
</math>
 +
* <math>\phi_3(x,y) =x^4+\dots,\quad \phi_4(x,y) = x^7+\dots</math>
 +
* interested in <math>F_m(x):=\phi_m(x,x)\in \Z[x]</math> :
 +
:<math>
 +
F_1(x)=0
 +
</math>
 +
:<math>
 +
F_2(x) = -(x-1728)(x+3375)^2(x-8000) = -H_{4}(x)H_{7}(x)^2H_{8}(x)
 +
</math>
 +
:<math>
 +
F_3(x) = -x(x-8000)^2  (x+32768)^2(x-54000)  = - H_3(x)H_{8}(x)^2H_{11}(x)^2H_{12}(x)
 +
</math>
 +
 
 +
* <math>F_m(x)\neq 0</math> if <math>m</math> is not a perfect square
 +
 
 +
* Hurwitz calculated its degree :
 +
:<math>\deg F_m(x)= \sum_{d|m}\max(d,m/d)</math>
 +
 
 +
* Kronecker : explicit factor. in class poly:
 +
:<math>
 +
F_m(x) =\pm \prod_{t\in \Z,t^2 \leq 4m}\mathcal{H}_{4m − t^2}(x)
 +
</math>
 +
where
 +
:<math>
 +
\mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}}
 +
</math>
 +
* can be also written as a product of <math>H_d(x)</math>
 +
 
 +
;thm (H.-K. class number relation)
 +
If <math>m</math> is not a perfect sq., then
 +
:<math>
 +
\sum_{d|m}\max(d,m/d) = \sum_{t\in \Z,t^2 \leq 4m}h_{4m − t^2}
 +
</math>
 +
 
 +
* this is surprising ; class numbers with different disc. have a linear relation!
 +
* geometric interpretation : <math>\deg F_m(x)</math> = number of intersections of two curves <math>\phi_1(x,y)=x-y=0</math> and <math>\phi_m(x,y)=0</math> in <math>\C^2</math>
 +
* Hurwitz computed this for pairs <math>\phi_{m_1}</math> and <math>\phi_{m_2}</math>
 +
 
 +
 
 +
;thm (Gross-Keating, 1993)
 +
<math>m_1,m_2,m_3</math> : pos int, and <math>A=\Z[X,Y]/\langle \phi_{m_1},\phi_{m_2},\phi_{m_3}\rangle</math>
 +
 
 +
# <math>\#A</math> is finite if and only if there is no pos. def. form <math>[a,b,c]</math> which represents <math>m_1,m_2,m_3</math>.
 +
# Let <math>\log \#A=\sum_{p}n(p)\log p</math>. Then <math>n(p)=0</math> for <math>p>4m_1m_2m_3</math>. For <math>p\leq 4m_1m_2m_3</math>,
 +
:<math>
 +
n(p) = \frac{1}{2}\sum_{Q}\left(\prod_{l\mid 4\det Q,\, l\neq p} \beta_l(Q) \right)\cdot \alpha_p(Q).
 +
</math>
 +
* sum is over all pos. def. half-int. <math>Q</math> over <math>\Z</math> with diagonal <math>(m_1,m_2,m_3)</math> which are isotropic over <math>\Q_{l}</math> for all <math>l\neq p</math> and anisotropic over <math>\Qp</math>
 +
* <math>\alpha_p(Q)</math> and <math>\beta_p(Q)</math> given in terms of <math>GK(Q)=(a_1,a_2,a_3)</math> (<math>Q</math> as a mat. over <math>\Qp</math>). For example,
 +
If <math>a_1\not\equiv a_2 \pmod 2</math>,
 +
:<math>
 +
\alpha_p(Q) = \sum_{i=0}^{a_1-1} (i+1) (a_1+a_2+a_3-3 i)p^i +\sum _{i=a_1}^{(a_1+a_2-1)/2} (a_1+1) (2a_1+a_2+a_3-4i)p^i.
 +
</math>
 +
:<math>
 +
\beta_p(Q) = \sum _{i=0}^{a_1-1} 2(i+1)p^i +\sum _{i=a_1}^{(a_1+a_2-2)/2} 2(a_1+1)p^i.
 +
</math>
 +
* => <math>\#A</math> :  arithmetic intersection number of divisors corr. to <math>\phi_m</math> on <math>S=\mathrm{Spec}\, \Z[X,Y]</math>
 +
 
 +
==repn of integers by quad. forms==
 +
* <math>Q</math> : a pos. def. quad. form <math>/\Z</math> in <math>n</math> var., i.e. <math>Q(X) = X^t A_{Q} X</math> for some pos. def. half-int. mat. <math>A_{Q}</math>, <math>X\in \Z^n</math>
 +
* <math>r(Q, m),\, m\geq 0</math> : number of <math>X\in \Z^n</math> such that <math>Q(X) = m</math>
 +
* theta function of <math>Q</math>
 +
:<math>
 +
\theta_Q(\tau)=\sum_{m=0}^\infty r(Q, m)q^{m}
 +
</math>
 +
* set <math>\det Q := \det (2A_Q)</math>
 +
* level <math>N</math> of <math>Q</math> : smallest int. <math>N</math> such that <math>N(2A_Q)^{-1}</math> is twice of a half-int mat.
 +
* for example, <math>Q=4x^2+6y^2</math>, <math>\det Q = 96</math>, <math>N=48</math>
 +
;thm  (see [[Theta function of a quadratic form]])
 +
For simplicity assume that <math>Q</math> has even number of var. (i.e. <math>n</math> even)
 +
 
 +
For <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)</math> with <math>c\equiv 0 \pmod N</math>,
 +
:<math>
 +
\theta_Q\left(\frac{a\tau+b}{c\tau+d}\right) = \left(\frac{(-1)^{n/2}\det Q}{d}\right)(c\tau+d)^{n/2}\theta_Q(\tau)
 +
</math>
 +
i.e., <math>\theta_Q</math> is a modular form of weight <math>n/2</math> with a Dirichlet character w.r.t. <math>\Gamma_0(N)</math>
 +
* space of modular forms with given weight, level, character  =  (space of Eisenstein series) + (space of cusp forms)
 +
* <math>\theta_Q(\tau) = E_Q(\tau)+C_Q(\tau)</math>
 +
* <math>r(Q, m)</math> = Fourier coef. of <math>E_Q(\tau)</math> + Fourier coef. of <math>C_Q(\tau)</math> (i.e. dominant term + error term)
 +
 
 +
===Siegel-Weil formula===
 +
* key message : single form : hard ; consider all forms in its genus
 +
* aut. gp. of <math>Q</math> : <math>{\rm Aut}(Q) = \{U\in GL_{n}(\Z):U^t A_Q U = A_Q\}</math>
 +
;def (genus of quad. form <math>/\Z</math>)
 +
<math>{\rm gen}(Q)</math> : set of <math>\Z</math>-equiv. classes of quad. forms that are <math>\Z_p</math>-equivalent to <math>Q</math> at all <math>p</math> (including <math>p=\infty</math>)
 +
 
 +
When <math>Q</math> is pos. def., <math>{\rm gen}(Q)</math> is finite (local-global fails)
 +
 
 +
;example (skip if no time)
 +
<math>f_1(x,y) =x^2+82y^2</math> and <math>f_2(x,y) =2x^2+41y^2</math> are <math>\Zp</math>-equivalent for all <math>p</math> , but not <math>\Z</math>-equivalent
 +
 
 +
 
 +
;thm (Siegel)
 +
<math>Q</math> : a pos. def. quad form <math>.\Z</math>. on <math>n</math> var.
 +
 
 +
To each <math>Q' \in {\rm gen}(Q)</math>, assign weight <math>w(Q')</math> proportional to <math>\frac{1}{|{\rm Aut}(Q')|}</math> so that <math>\sum_{Q'} w(Q')=1</math> i.e.
 +
:<math>
 +
w(Q') = \frac{1}{|{\rm Aut}(Q')|}\,\cdot\,\left(\sum_{Q'\in {\rm gen}(Q)}\frac{1}{|{\rm Aut}(Q')|}\right)^{-1}
 +
</math>
 +
# weighted average of theta functions : :<math>\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)</math>
 +
# weighted average of representation number (i.e. Fourier coef. of <math>E_Q</math>)
 +
:<math>
 +
\sum_{Q'\in {\rm gen}(Q)}w(Q')r(Q', m)=(\text{const. on }n) \prod_{p:\text{primes}}\alpha_{p}(Q,m) = (*) \alpha_{\infty}(Q,m)\alpha_{2}(Q,m)\alpha_{3}(Q,m)\dots
 +
</math>
 +
where <math>\alpha_{p}(Q,m)</math> is local density at <math>p</math> (will be defined soon).
 +
;remark
 +
* regard <math>m\in \Z_{\geq 0}</math> as half-int. <math>1\times 1</math> mat
 +
* <math>A</math> and <math>B</math> be half-int. over <math>\Z</math> of size <math>m</math> and <math>n</math>, <math>m\geq n\geq 1</math>
 +
* <math>r(A,B)</math> : number of <math>m \times n</math> int. mat. <math>X</math> s.t. <math>X^t A X = B</math>
 +
* Siegel's theorem holds for <math>r(A,B)</math>, modular form becomes Siegel modular forms
 +
 
 +
===Local density and Siegel series===
 +
;def (local density)
 +
Define
 +
:<math>
 +
\alpha_{p}(A,B)= \lim_{\ell\to\infty}p^{-\ell(mn-n(n+1)/2)}N_{p^{\ell}}(A,B)
 +
</math>
 +
where
 +
:<math>
 +
N_{p^{\ell}}(A,B) = \#\{X\in M_{m\times n}(\Zp/p^{\ell}\Zp)\, | X^{t}AX = B \pmod{p^{\ell}\calh_n(\Zp)}\}
 +
</math>
 +
* <math>\alpha_{p}(A,B)</math> : very difficult to compute in general
 +
* <math>\exists</math> important special case we know more
 +
 
 +
;thm (?Kitaoka)
 +
<math>B\in \matn</math>. <math>\exists</math> a poly <math>f_p(B;X)\in \Z[X]</math> such that for <math>k\geq n</math>,
 +
:<math>
 +
f_p(B;p^{-k}) = \alpha_{p}(H_{k},B)
 +
</math>
 +
where <math>H_k=\underbrace{\left(
 +
\begin{array}{cc}
 +
0 & \frac{1}{2} \\
 +
\frac{1}{2} & 0 \\
 +
\end{array}
 +
\right)\bot \dots \bot \left(
 +
\begin{array}{cc}
 +
0 & \frac{1}{2} \\
 +
\frac{1}{2} & 0 \\
 +
\end{array}
 +
\right)}_{k}</math>
 +
 
 +
;def
 +
Siegel series of <math>B</math> : <math>f_p(B;X)</math> (more precisely, <math>f_p(B;p^{-s}),\, s\in \C</math>)
 +
 
 +
;remark
 +
* Siegel series : <math>p</math>-local factor of Fourier coef. of Siegel-Eisenstein series (for <math>\operatorname{Sp}_{n}(\Z)</math>, or weighted average for even unimodular lattices)
 +
 
 +
;thm (Ikeda-Katsurada 2016)
 +
Siegel series of <math>B</math> only depends on <math>GK(B)=(a_1,\dots, a_n)</math> (there is an algorithm to compute it from <math>GK(B)</math>)
 +
 
 +
==memo==
 +
* Eisenstein series
 +
:<math>
 +
E_{2k}(\tau)=1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)
 +
</math>
 +
:<math>
 +
E_{12}(\tau) =1+ \frac{65520 q}{691}+\frac{134250480 q^2}{691}+\dots
 +
</math>
 +
 
 +
===Siegel modular forms===
 +
A Siegel modular form <math>f</math> of genus <math>g</math> has an expansion of the form
 +
:<math>f(Z)=\sum_{T\in \Xgsemi}a(T;f)\e(\ip TZ)</math>
 +
where <math>\e(\ip TZ):=\exp\left(2\pi i \operatorname{Tr}(TZ)\right)</math> and <math>\Xgsemi</math> denotes the set of half-int. pos. semi-def symm.
 +
<math>g\times g</math> matrices.
 +
}
 +
 
 +
;example Fourier expansion in genus 2
 +
Let <math>f</math> be a Siegel modular form of genus 2 and consider its Fourier expansion
 +
:<math>f(Z)=\sum_{T\in \Xtwo}a(T;f)\e(\ip TZ).</math>
 +
 
 +
For
 +
<math>
 +
T=\begin{pmatrix}a & b/2 \\ b/2 & c \\\end{pmatrix} \in \Xtwo
 +
</math>
 +
and
 +
<math>
 +
Z=\begin{pmatrix}\tau_1 & z \\ z & \tau_2 \\\end{pmatrix}\in \hh{2}
 +
</math>,
 +
:<math>
 +
\operatorname{Tr}(T Z)=a \tau_1+b z+c \tau_2.
 +
</math>
 +
 
 +
If we set <math>q_i=e^{2\pi i \tau_i}</math>, <math>\zeta=e^{2\pi i z}</math>, then
 +
:<math>
 +
\e(\ip TZ)=\exp\left(2\pi i \operatorname{Tr}(T Z)\right)=q_1^a\zeta^bq_2^c
 +
</math>
 +
and thus,
 +
:<math>f(Z)=\sum_{T\in \Xtwo}a(T;f)q_1^a\zeta^bq_2^c.</math>
 +
 
 +
===Fourier coef.s of Siegel-Eisenstein series===
 +
The Eisenstein series of weight <math>k</math> (even) and genus <math>g</math> is
 +
:<math>
 +
\Egk(Z) = \sum_{\tiny{\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}}\in \Gamma_{g,0}\backslash \Gamma_{g}} \frac{1}{\det(CZ +D)^{k}},
 +
</math>
 +
where
 +
:<math>
 +
\Gamma_{g,0}=\{\begin{pmatrix}A & B \\ 0 & D \\\end{pmatrix}\in \Gamma_{g}\}.
 +
</math>
 +
In other words, the summation is over all classes of coprime pairs <math>(C,D)</math>.
 +
 
 +
The Eisenstein series <math>\Egk(Z)</math> is a Siegel modular form of weight <math>k</math> and of genus <math>g</math>.
  
−
Moreover, the Fourier coefficients $a_{E}(N)$ of $E$ can be expressed as an infinite product of [[Local density of quadratic form|local densities]]
+
Consider the Fourier expansion of <math>\Egk(Z)</math> :
−
$$
+
:<math>
−
a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp}
+
\Egk(Z)=\sum_{T\in\Xgsemi}\fc T{\Egk}\,\e(\ip TZ).
−
$$
+
</math>
−
===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
 
  
 +
;thm (Kitaoka?)
 +
'''check the condition on <math>k</math> and <math>g</math> for the formula'''
 +
 +
Assume that <math>k>g</math>. For non-deg. <math>T\in\Xgsemi</math>,
 +
:<math>
 +
\fc T\Egk=
 +
\dfrac{2^{\lfloor \frac{g+1}{2} \rfloor} \prod_{p}F_p(T,p^{k-g-1})}
 +
{\zeta(1-k)\prod_{i=1}^{\lfloor g/2\rfloor}\zeta(1-2k+2i)}
 +
\cdot\begin{cases}
 +
L_{D_T}(1-k+g/2)&\text{$g$ even},\\
 +
1&\text{$g$ odd}
 +
\end{cases}
 +
</math>
 +
where <math>F_p(T,X)\in \Z[X]</math> depending only on the <math>\Zp</math>-class of <math>T</math>. The product is over all primes <math>p\mid2\det(2T)</math>.
  
 
==related items==
 
==related items==
 +
* [[Talk on Siegel theta series and modular forms]]
 +
* [[Talk on Fourier coefficients of Siegel-Eisenstein series]]
 +
* [[Fourier coefficients of Siegel-Eisenstein series]]
 
* [[Gross-Keating invariants of a quadratic form]]
 
* [[Gross-Keating invariants of a quadratic form]]
 
* [[Siegel-Weil formula]]
 
* [[Siegel-Weil formula]]
39번째 줄: 387번째 줄:
 
* {{수학노트||url=J-불변량과_모듈라_다항식}}
 
* {{수학노트||url=J-불변량과_모듈라_다항식}}
 
* {{수학노트|url=지겔-베유_공식}}
 
* {{수학노트|url=지겔-베유_공식}}
 +
* {{수학노트|url=후르비츠-크로네커_유수}}
 +
* {{수학노트|url=타원 모듈라 j-함수의 singular moduli}}
 +
 +
==computational resource==
 +
* https://drive.google.com/file/d/1kj7C_5xBW8nGMQiOOAc31nXTfAeP8ykG/view
  
  
 
[[분류:talks and lecture notes]]
 
[[분류:talks and lecture notes]]
 +
[[분류:theta]]
 +
[[분류:migrate]]

2020년 11월 16일 (월) 08:38 기준 최신판

overview

  • defn of Gross-Keating inv. of a quad. form over Zp
  • bin. quad. forms and class number relations
  • representation of integers by quad. forms
  • (skip if no time) computer program that computes Gross-Keating inv. of a quad. form over Zp

\( \newcommand{\Z}{\mathbb Z} \newcommand{\Zn}{\Z_{\geq 0}^n} \newcommand{\Zp}{\mathbb {Z}_p} \newcommand{\matn}{\calh_n(\Zp)^{\rm nd}} \def\Zmat#1{\calh_{#1}(\Z)^{\rm nd}} \def\mat#1#2{\calh_{#1}(\Z_{#2})^{\rm nd}} \newcommand{\ord}{\mathrm{ord}} \newcommand{\calh}{\mathcal H} \newcommand{\frko}{\mathfrak o} \newcommand{\GL}{{\mathrm{GL}}} \newcommand{\GK}{\mathrm{GK}} \newcommand{\vep}{\varepsilon} \newcommand{\intmult}{(T_{m_1} \cdot T_{m_2}\cdot T_{m_3})_{S}} \newcommand{\Qp}{\mathbb {Q}_p} \newcommand{\diag}{\mathrm{diag}} \def\sym#1{{\rm Sym}_n(#1)} \newcommand\supparen[1]{^{(#1)}} \newcommand\Egk{E_k\supparen g} \newcommand\GLnZ{\GL n\Z} \newcommand\Xgsemi{\siX g^{\rm semi}} \newcommand\fc[2]{a(#1;#2)} \newcommand\e{\operatorname{e}} \newcommand\ip[2]{\langle #1,#2\rangle} \newcommand\siX[1]{{\mathcal X}_{#1}} \newcommand\Xn{\siX n} \newcommand\Xm{\siX m} \newcommand\Xg{\siX g} \newcommand\Xnsemi{\siX n^{\rm semi}} \newcommand\Xgsemi{\siX g^{\rm semi}} \newcommand\Xtwo{\siX 2^{\rm semi}} \newcommand\hh[1]{\mathbb{H}_{#1}} \)

Gross-Keating inv.

  • Gross-Keating invariants of a quadratic form
  • \(p\in \Z_{> 0}\) : prime
  • \(\Qp\) \[p\]-adic completion of \(\Q\), and \(\Zp\) : ring of int.
  • For \(a\in \Qp^\times\), \(\ord(a)=n\) if \(a\in p^n \Zp^\times\), \(\ord(0)=\infty\)
  • symm. \(n\times n\) mat. \(B=(b_{ij}),\, b_{ij}\in \Qp\) is half-integral if \(b_{ii}\in \Zp\) and \(2b_{ij}\in \Zp\)
  • \(\matn\) : set of \(n\times n\) non-deg. half-int. mat.
def

\(B=(b_{ij})\in\matn\)

\(S(B)\) : set of all non-decreasing seq. \((a_1, \dots, a_n)\in\Zn\) s.t. \begin{align*} &\ord(b_{ii})\geq a_i \qquad\qquad\qquad\quad (1\leq i\leq n), \\ &\ord(2 b_{ij})\geq (a_i+a_j)/2 \qquad\; (1\leq i\leq j\leq n), \end{align*}

\(S(\{B\}):=\bigcup_{U\in\GL_n(\Zp)} S(U^{t}BU)\)

GK inv. \(\GK(B)=(a_1, \dots, a_n)\in\Zn\) of \(B\) is \begin{align*} a_1&=\max_{(y_1, \dots)\in S(\{B\})} \,y_1, \\ a_2&=\max_{(a_1, y_2, \dots)\in S(\{B\})}\, y_2, \\ &\dots \\ a_n&=\max_{(a_1, a_2, \dots, a_{n-1}, y_n)\in S(\{B\})}\, y_n. \end{align*}

  • By definition \(GK(B)\) depends only on \(\Zp\)-class of \(B\) under \(B\sim B'\) if \(B' = U^{t}BU\) for some \(U\in\GL_n(\Zp)\)
  • hard to compute from definition
remarks
  • 1993 : Gross-Keating : introduced \(GK(B)\) for 3x3 \(B\) in study of arithmetic intersection number related to three modular poly.
  • 2015 : Ikeda-Katsurada : defined \(GK(B)\) for \(B\) \(n\times n\) half-int. over a finite ext'n of \(\Qp\)
  • 2016 : Ikeda-Katsurada : Siegel series of \(B\) (local factor of Fourier coef of Siegel-Eisenstein series) is determined by \(GK(B)\)
  • 2017 : Cho-Ikeda-Katsurada-Yamauchi : computer-friendly (not human-friendly) inductive formulas for \(GK(B)\)
  • I recently wrote computer program using Mathematica; arXiv:1809.10323

bin. quad. forms and class number relations

  • \(Q=Ax^2+Bxy+Cy^2\) : pos. def. bin. quad. form over \(\Z\), write \(Q=[A,B,C]\)
  • disc. of \(Q\) \[B^2-4AC<0\]
  • for int. \(d>0\),
    • \(\mathcal{Q}_d=\{Q:B^2-4AC=-d\}\)
    • \(\mathcal{Q}_{d}^{pr}=\{Q\in \mathcal{Q}_d:\text{primitive}\}\). \(Q\) is prim. if \(\rm{GCD}(A,B,C)=1\)
  • \(\Gamma=PSL_2(\mathbb{Z})\) acts on \(\mathcal{Q}_d\) (and \(\mathcal{Q}_{d}^{pr}\)) \[Q\mapsto Q'\] by \(Q'(x,y)=Q(ax+by,cx+dy)\)

\[ \left( \begin{array}{cc} A & \frac{B}{2} \\ \frac{B}{2} & C \\ \end{array} \right) \mapsto \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)^t\left( \begin{array}{cc} A & \frac{B}{2} \\ \frac{B}{2} & C \\ \end{array} \right)\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \]

  • for each \(Q\), \(w_{Q}\) : size of stabilizers
    • \(w_Q=2\) if \(Q\sim [a,0,a]\)
    • \(w_Q=3\) if \(Q\sim [a,a,a]\)
    • \(w_Q=1\) otherwise
def (class number and Hurwitz-Kronecker class number)

For int. \(d>0\), \[h_{d}^{pr}:=\#(\mathcal{Q}_d^{pr}/\Gamma),\quad h_d:=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}\]

example
  • \(\mathcal{Q}_{12}^{pr}/\Gamma = \{[1,0,3]\}\), \(h_{12}^{pr} = 1\)
  • \(\mathcal{Q}_{12}/\Gamma = \{[1,0,3],[2,2,2]\}\), \(h_{12} = 4/3\)
  • when \(d=23\), both are \[\{[1,1,6], [2,-1,3], [2,1,3]\}\], \(h_{23}=h_{23}^{pr} = 3\)


class poly

def (j-inv.)

\[ j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+\cdots,\, q=e^{2\pi i\tau},\tau\in \mathbb{H} \] where \[ E_ 4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n,\quad \sigma_3(n)=\sum_{d|n}d^3\] \[\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}\]


thm

\(Q=[A,B,C]\) : prim of disc \(-d\), and \(\tau_Q = \frac{-B+\sqrt{B^2-4AC}}{2A}\in \mathbb{H}\).

Then \(j(\tau_Q)\) is an alg. int. with minimal poly. \[ H_d(x) : = \prod_{Q\in \mathcal{Q}_{d}^{\rm{pr}}/\Gamma}(x-j(\tau_Q))\in \Z[x] \] In particular, \(h_{d}^{\rm{pr}}=1\), then \(j(\tau_Q)\in \mathbb{Z}\).

example

\(h_{23}^{pr}=3, \qquad H_{23}(x) = x^3+3491750 x^2-5151296875 x+12771880859375\)

modular poly

  • \(m>0\) : int
  • \(\exists\) \(\phi_m(x,y)\in{\mathbb{Z}}[x,y]\) such that

\[\prod_{ad=m,a,d>0,0\leq b \leq d-1}(x-j(\frac{a\tau+b}{d}))=\phi_m(x,j(\tau))\]

  • \(\phi_m(j(m\tau),j(\tau))=0\)
  • as a poly. in \(x\), \(\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d\)
examples
  • \(m=1\), \(\phi_1(x,y)=x-y\)
  • \(m=2\)

\[ \phi_2(x,y)=x^3+y^3-x^2 y^2+1488 (x^2 y + x y^2)-162000 (x^2+y^2) +40773375 x y+8748000000 (x + y)-157464000000000 \]

  • \(\phi_3(x,y) =x^4+\dots,\quad \phi_4(x,y) = x^7+\dots\)
  • interested in \(F_m(x):=\phi_m(x,x)\in \Z[x]\) :

\[ F_1(x)=0 \] \[ F_2(x) = -(x-1728)(x+3375)^2(x-8000) = -H_{4}(x)H_{7}(x)^2H_{8}(x) \] \[ F_3(x) = -x(x-8000)^2 (x+32768)^2(x-54000) = - H_3(x)H_{8}(x)^2H_{11}(x)^2H_{12}(x) \]

  • \(F_m(x)\neq 0\) if \(m\) is not a perfect square
  • Hurwitz calculated its degree :

\[\deg F_m(x)= \sum_{d|m}\max(d,m/d)\]

  • Kronecker : explicit factor. in class poly:

\[ F_m(x) =\pm \prod_{t\in \Z,t^2 \leq 4m}\mathcal{H}_{4m − t^2}(x) \] where \[ \mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}} \]

  • can be also written as a product of \(H_d(x)\)
thm (H.-K. class number relation)

If \(m\) is not a perfect sq., then \[ \sum_{d|m}\max(d,m/d) = \sum_{t\in \Z,t^2 \leq 4m}h_{4m − t^2} \]

  • this is surprising ; class numbers with different disc. have a linear relation!
  • geometric interpretation \[\deg F_m(x)\] = number of intersections of two curves \(\phi_1(x,y)=x-y=0\) and \(\phi_m(x,y)=0\) in \(\C^2\)
  • Hurwitz computed this for pairs \(\phi_{m_1}\) and \(\phi_{m_2}\)


thm (Gross-Keating, 1993)

\(m_1,m_2,m_3\) : pos int, and \(A=\Z[X,Y]/\langle \phi_{m_1},\phi_{m_2},\phi_{m_3}\rangle\)

  1. \(\#A\) is finite if and only if there is no pos. def. form \([a,b,c]\) which represents \(m_1,m_2,m_3\).
  2. Let \(\log \#A=\sum_{p}n(p)\log p\). Then \(n(p)=0\) for \(p>4m_1m_2m_3\). For \(p\leq 4m_1m_2m_3\),

\[ n(p) = \frac{1}{2}\sum_{Q}\left(\prod_{l\mid 4\det Q,\, l\neq p} \beta_l(Q) \right)\cdot \alpha_p(Q). \]

  • sum is over all pos. def. half-int. \(Q\) over \(\Z\) with diagonal \((m_1,m_2,m_3)\) which are isotropic over \(\Q_{l}\) for all \(l\neq p\) and anisotropic over \(\Qp\)
  • \(\alpha_p(Q)\) and \(\beta_p(Q)\) given in terms of \(GK(Q)=(a_1,a_2,a_3)\) (\(Q\) as a mat. over \(\Qp\)). For example,

If \(a_1\not\equiv a_2 \pmod 2\), \[ \alpha_p(Q) = \sum_{i=0}^{a_1-1} (i+1) (a_1+a_2+a_3-3 i)p^i +\sum _{i=a_1}^{(a_1+a_2-1)/2} (a_1+1) (2a_1+a_2+a_3-4i)p^i. \] \[ \beta_p(Q) = \sum _{i=0}^{a_1-1} 2(i+1)p^i +\sum _{i=a_1}^{(a_1+a_2-2)/2} 2(a_1+1)p^i. \]

  • => \(\#A\) : arithmetic intersection number of divisors corr. to \(\phi_m\) on \(S=\mathrm{Spec}\, \Z[X,Y]\)

repn of integers by quad. forms

  • \(Q\) : a pos. def. quad. form \(/\Z\) in \(n\) var., i.e. \(Q(X) = X^t A_{Q} X\) for some pos. def. half-int. mat. \(A_{Q}\), \(X\in \Z^n\)
  • \(r(Q, m),\, m\geq 0\) : number of \(X\in \Z^n\) such that \(Q(X) = m\)
  • theta function of \(Q\)

\[ \theta_Q(\tau)=\sum_{m=0}^\infty r(Q, m)q^{m} \]

  • set \(\det Q := \det (2A_Q)\)
  • level \(N\) of \(Q\) : smallest int. \(N\) such that \(N(2A_Q)^{-1}\) is twice of a half-int mat.
  • for example, \(Q=4x^2+6y^2\), \(\det Q = 96\), \(N=48\)
thm (see Theta function of a quadratic form)

For simplicity assume that \(Q\) has even number of var. (i.e. \(n\) even)

For \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\Z)\) with \(c\equiv 0 \pmod N\), \[ \theta_Q\left(\frac{a\tau+b}{c\tau+d}\right) = \left(\frac{(-1)^{n/2}\det Q}{d}\right)(c\tau+d)^{n/2}\theta_Q(\tau) \] i.e., \(\theta_Q\) is a modular form of weight \(n/2\) with a Dirichlet character w.r.t. \(\Gamma_0(N)\)

  • space of modular forms with given weight, level, character = (space of Eisenstein series) + (space of cusp forms)
  • \(\theta_Q(\tau) = E_Q(\tau)+C_Q(\tau)\)
  • \(r(Q, m)\) = Fourier coef. of \(E_Q(\tau)\) + Fourier coef. of \(C_Q(\tau)\) (i.e. dominant term + error term)

Siegel-Weil formula

  • key message : single form : hard ; consider all forms in its genus
  • aut. gp. of \(Q\) \[{\rm Aut}(Q) = \{U\in GL_{n}(\Z):U^t A_Q U = A_Q\}\]
def (genus of quad. form \(/\Z\))

\({\rm gen}(Q)\) : set of \(\Z\)-equiv. classes of quad. forms that are \(\Z_p\)-equivalent to \(Q\) at all \(p\) (including \(p=\infty\))

When \(Q\) is pos. def., \({\rm gen}(Q)\) is finite (local-global fails)

example (skip if no time)

\(f_1(x,y) =x^2+82y^2\) and \(f_2(x,y) =2x^2+41y^2\) are \(\Zp\)-equivalent for all \(p\) , but not \(\Z\)-equivalent


thm (Siegel)

\(Q\) : a pos. def. quad form \(.\Z\). on \(n\) var.

To each \(Q' \in {\rm gen}(Q)\), assign weight \(w(Q')\) proportional to \(\frac{1}{|{\rm Aut}(Q')|}\) so that \(\sum_{Q'} w(Q')=1\) i.e. \[ w(Q') = \frac{1}{|{\rm Aut}(Q')|}\,\cdot\,\left(\sum_{Q'\in {\rm gen}(Q)}\frac{1}{|{\rm Aut}(Q')|}\right)^{-1} \]

  1. weighted average of theta functions : \[\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)\]
  2. weighted average of representation number (i.e. Fourier coef. of \(E_Q\))

\[ \sum_{Q'\in {\rm gen}(Q)}w(Q')r(Q', m)=(\text{const. on }n) \prod_{p:\text{primes}}\alpha_{p}(Q,m) = (*) \alpha_{\infty}(Q,m)\alpha_{2}(Q,m)\alpha_{3}(Q,m)\dots \] where \(\alpha_{p}(Q,m)\) is local density at \(p\) (will be defined soon).

remark
  • regard \(m\in \Z_{\geq 0}\) as half-int. \(1\times 1\) mat
  • \(A\) and \(B\) be half-int. over \(\Z\) of size \(m\) and \(n\), \(m\geq n\geq 1\)
  • \(r(A,B)\) : number of \(m \times n\) int. mat. \(X\) s.t. \(X^t A X = B\)
  • Siegel's theorem holds for \(r(A,B)\), modular form becomes Siegel modular forms

Local density and Siegel series

def (local density)

Define \[ \alpha_{p}(A,B)= \lim_{\ell\to\infty}p^{-\ell(mn-n(n+1)/2)}N_{p^{\ell}}(A,B) \] where \[ N_{p^{\ell}}(A,B) = \#\{X\in M_{m\times n}(\Zp/p^{\ell}\Zp)\, | X^{t}AX = B \pmod{p^{\ell}\calh_n(\Zp)}\} \]

  • \(\alpha_{p}(A,B)\) : very difficult to compute in general
  • \(\exists\) important special case we know more
thm (?Kitaoka)

\(B\in \matn\). \(\exists\) a poly \(f_p(B;X)\in \Z[X]\) such that for \(k\geq n\), \[ f_p(B;p^{-k}) = \alpha_{p}(H_{k},B) \] where \(H_k=\underbrace{\left( \begin{array}{cc} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \\ \end{array} \right)\bot \dots \bot \left( \begin{array}{cc} 0 & \frac{1}{2} \\ \frac{1}{2} & 0 \\ \end{array} \right)}_{k}\)

def

Siegel series of \(B\) \[f_p(B;X)\] (more precisely, \(f_p(B;p^{-s}),\, s\in \C\))

remark
  • Siegel series \[p\]-local factor of Fourier coef. of Siegel-Eisenstein series (for \(\operatorname{Sp}_{n}(\Z)\), or weighted average for even unimodular lattices)
thm (Ikeda-Katsurada 2016)

Siegel series of \(B\) only depends on \(GK(B)=(a_1,\dots, a_n)\) (there is an algorithm to compute it from \(GK(B)\))

memo

  • Eisenstein series

\[ E_{2k}(\tau)=1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right) \] \[ E_{12}(\tau) =1+ \frac{65520 q}{691}+\frac{134250480 q^2}{691}+\dots \]

Siegel modular forms

A Siegel modular form \(f\) of genus \(g\) has an expansion of the form \[f(Z)=\sum_{T\in \Xgsemi}a(T;f)\e(\ip TZ)\] where \(\e(\ip TZ):=\exp\left(2\pi i \operatorname{Tr}(TZ)\right)\) and \(\Xgsemi\) denotes the set of half-int. pos. semi-def symm. \(g\times g\) matrices. }

example Fourier expansion in genus 2

Let \(f\) be a Siegel modular form of genus 2 and consider its Fourier expansion \[f(Z)=\sum_{T\in \Xtwo}a(T;f)\e(\ip TZ).\]

For \( T=\begin{pmatrix}a & b/2 \\ b/2 & c \\\end{pmatrix} \in \Xtwo \) and \( Z=\begin{pmatrix}\tau_1 & z \\ z & \tau_2 \\\end{pmatrix}\in \hh{2} \), \[ \operatorname{Tr}(T Z)=a \tau_1+b z+c \tau_2. \]

If we set \(q_i=e^{2\pi i \tau_i}\), \(\zeta=e^{2\pi i z}\), then \[ \e(\ip TZ)=\exp\left(2\pi i \operatorname{Tr}(T Z)\right)=q_1^a\zeta^bq_2^c \] and thus, \[f(Z)=\sum_{T\in \Xtwo}a(T;f)q_1^a\zeta^bq_2^c.\]

Fourier coef.s of Siegel-Eisenstein series

The Eisenstein series of weight \(k\) (even) and genus \(g\) is \[ \Egk(Z) = \sum_{\tiny{\begin{pmatrix}A & B \\ C & D \\\end{pmatrix}}\in \Gamma_{g,0}\backslash \Gamma_{g}} \frac{1}{\det(CZ +D)^{k}}, \] where \[ \Gamma_{g,0}=\{\begin{pmatrix}A & B \\ 0 & D \\\end{pmatrix}\in \Gamma_{g}\}. \] In other words, the summation is over all classes of coprime pairs \((C,D)\).

The Eisenstein series \(\Egk(Z)\) is a Siegel modular form of weight \(k\) and of genus \(g\).

Consider the Fourier expansion of \(\Egk(Z)\) : \[ \Egk(Z)=\sum_{T\in\Xgsemi}\fc T{\Egk}\,\e(\ip TZ). \]

thm (Kitaoka?)

check the condition on \(k\) and \(g\) for the formula

Assume that \(k>g\). For non-deg. \(T\in\Xgsemi\), \[ \fc T\Egk= \dfrac{2^{\lfloor \frac{g+1}{2} \rfloor} \prod_{p}F_p(T,p^{k-g-1})} {\zeta(1-k)\prod_{i=1}^{\lfloor g/2\rfloor}\zeta(1-2k+2i)} \cdot\begin{cases} L_{D_T}(1-k+g/2)&\text{$g$ even},\\ 1&\text{$g$ odd} \end{cases} \] where \(F_p(T,X)\in \Z[X]\) depending only on the \(\Zp\)-class of \(T\). The product is over all primes \(p\mid2\det(2T)\).

related items

computational resource

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