"Talk on Gross-Keating invariants"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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(사용자 2명의 중간 판 275개는 보이지 않습니다) | |||
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==overview== | ==overview== | ||
− | * | + | * defn of Gross-Keating inv. of a quad. form over Zp |
− | * | + | * bin. quad. forms and class number relations |
− | * representation of integers by | + | * 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{\Z}{\mathbb Z} | ||
\newcommand{\Zn}{\Z_{\geq 0}^n} | \newcommand{\Zn}{\Z_{\geq 0}^n} | ||
\newcommand{\Zp}{\mathbb {Z}_p} | \newcommand{\Zp}{\mathbb {Z}_p} | ||
− | \newcommand{\matn}{\calh_n(\ | + | \newcommand{\matn}{\calh_n(\Zp)^{\rm nd}} |
\def\Zmat#1{\calh_{#1}(\Z)^{\rm nd}} | \def\Zmat#1{\calh_{#1}(\Z)^{\rm nd}} | ||
\def\mat#1#2{\calh_{#1}(\Z_{#2})^{\rm nd}} | \def\mat#1#2{\calh_{#1}(\Z_{#2})^{\rm nd}} | ||
38번째 줄: | 36번째 줄: | ||
\newcommand\Xtwo{\siX 2^{\rm semi}} | \newcommand\Xtwo{\siX 2^{\rm semi}} | ||
\newcommand\hh[1]{\mathbb{H}_{#1}} | \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*} | \begin{align*} | ||
&\ord(b_{ii})\geq a_i \qquad\qquad\qquad\quad (1\leq i\leq n), \\ | &\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), | &\ord(2 b_{ij})\geq (a_i+a_j)/2 \qquad\; (1\leq i\leq j\leq n), | ||
\end{align*} | \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*} | \begin{align*} | ||
a_1&=\max_{(y_1, \dots)\in S(\{B\})} \,y_1, \\ | a_1&=\max_{(y_1, \dots)\in S(\{B\})} \,y_1, \\ | ||
66번째 줄: | 65번째 줄: | ||
\end{align*} | \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> | :<math> | ||
− | j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+ | + | 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> | </math> | ||
where | where | ||
− | :<math> E_ 4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n | + | :<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>\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}</math> |
+ | |||
+ | |||
+ | ;thm | ||
+ | <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>. | ||
+ | |||
+ | Then <math>j(\tau_Q)</math> is an alg. int. with minimal poly. | ||
+ | :<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> |
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+ | * 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 | |
− | Z | + | :<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> | ||
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− | + | ;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=== | ===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== | + | ===Siegel modular forms=== |
− | A Siegel modular form | + | 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 | + | 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 | + | 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 | For | ||
− | + | <math> | |
T=\begin{pmatrix}a & b/2 \\ b/2 & c \\\end{pmatrix} \in \Xtwo | T=\begin{pmatrix}a & b/2 \\ b/2 & c \\\end{pmatrix} \in \Xtwo | ||
− | + | </math> | |
and | and | ||
− | + | <math> | |
Z=\begin{pmatrix}\tau_1 & z \\ z & \tau_2 \\\end{pmatrix}\in \hh{2} | 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. | \operatorname{Tr}(T Z)=a \tau_1+b z+c \tau_2. | ||
− | + | </math> | |
− | If we set | + | 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 | \e(\ip TZ)=\exp\left(2\pi i \operatorname{Tr}(T Z)\right)=q_1^a\zeta^bq_2^c | ||
− | + | </math> | |
and thus, | 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=== | |
− | ===Fourier | + | The Eisenstein series of weight <math>k</math> (even) and genus <math>g</math> is |
− | The Eisenstein series of weight | + | :<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}}, | \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 | where | ||
− | + | :<math> | |
\Gamma_{g,0}=\{\begin{pmatrix}A & B \\ 0 & D \\\end{pmatrix}\in \Gamma_{g}\}. | \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 | + | In other words, the summation is over all classes of coprime pairs <math>(C,D)</math>. |
− | |||
− | The Eisenstein series | + | The Eisenstein series <math>\Egk(Z)</math> is a Siegel modular form of weight <math>k</math> and of genus <math>g</math>. |
− | |||
− | + | Consider the Fourier expansion of <math>\Egk(Z)</math> : | |
− | + | :<math> | |
− | Consider the Fourier expansion of | ||
− | |||
\Egk(Z)=\sum_{T\in\Xgsemi}\fc T{\Egk}\,\e(\ip TZ). | \Egk(Z)=\sum_{T\in\Xgsemi}\fc T{\Egk}\,\e(\ip TZ). | ||
− | + | </math> | |
;thm (Kitaoka?) | ;thm (Kitaoka?) | ||
− | Assume that | + | '''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= | \fc T\Egk= | ||
\dfrac{2^{\lfloor \frac{g+1}{2} \rfloor} \prod_{p}F_p(T,p^{k-g-1})} | \dfrac{2^{\lfloor \frac{g+1}{2} \rfloor} \prod_{p}F_p(T,p^{k-g-1})} | ||
229번째 줄: | 374번째 줄: | ||
1&\text{$g$ odd} | 1&\text{$g$ odd} | ||
\end{cases} | \end{cases} | ||
− | + | </math> | |
− | where | + | 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>. |
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==related items== | ==related items== | ||
* [[Talk on Siegel theta series and modular forms]] | * [[Talk on Siegel theta series and modular forms]] | ||
+ | * [[Talk on Fourier coefficients of Siegel-Eisenstein series]] | ||
* [[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]] | ||
286번째 줄: | 389번째 줄: | ||
* {{수학노트|url=후르비츠-크로네커_유수}} | * {{수학노트|url=후르비츠-크로네커_유수}} | ||
* {{수학노트|url=타원 모듈라 j-함수의 singular moduli}} | * {{수학노트|url=타원 모듈라 j-함수의 singular moduli}} | ||
− | |||
==computational resource== | ==computational resource== | ||
294번째 줄: | 396번째 줄: | ||
[[분류:talks and lecture notes]] | [[분류:talks and lecture notes]] | ||
[[분류:theta]] | [[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 \(A=\Z[X,Y]/\langle \phi_{m_1},\phi_{m_2},\phi_{m_3}\rangle\)
- \(\#A\) is finite if and only if there is no pos. def. form \([a,b,c]\) which represents \(m_1,m_2,m_3\).
- 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. \[ w(Q') = \frac{1}{|{\rm Aut}(Q')|}\,\cdot\,\left(\sum_{Q'\in {\rm gen}(Q)}\frac{1}{|{\rm Aut}(Q')|}\right)^{-1} \]
- weighted average of theta functions : \[\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)\]
- 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 \(p\mid2\det(2T)\).
- 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
- Siegel-Weil formula
- Local density of quadratic form
- Local Siegel series and Katsurada Fp polynomial
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