Talk on Gross-Keating invariants
overview
- definition of the Gross-Keating invariant of a quadratic form over Zp
- binary quadratic forms and class number relations
- representation of integers by quadratic forms
- a computer program that computes the Gross-Keating invariant of a quadratic form over Zp
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Gross-Keating invariant
- Let $p\in \Z_{\geq 0}$ be a prime,
- $\Qp$ : $p$-adic completion of $\Q$, and $\Zp$ : ring of integers
- For $a\in \Qp^\times$, $\ord(a)=n$ if $a\in p^n \Zp^\times$, $\ord(0)=\infty$
- symmetric matrix $B=(b_{ij}),\, b_{ij}\in \Qp$ is half-integral if $b_{ii}\in \Zp$ and $2b_{ij}\in \Zp$
- $\matn$ : set of non-degenerate half-integral symmetric matrix of size $n$
- def (Gross-Keating invariant)
Let $B=(b_{ij})\in\matn$. Let $S(B)$ be the set of all non-decreasing sequences $(a_1, \dots, a_n)\in\Zn$ such that \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*} and $S(\{B\}):=\bigcup_{U\in\GL_n(\Zp)} S(U^{t}BU)$. The Gross-Keating invariant $\GK(B)=(a_1, \dots, a_n)\in\Zn$ of $B$ is defined by \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 $GK(B)$ only depends on $\Zp$-class of $B$ under the relation $B\sim B'$ if $B' = U^{t}BU$ for some $U\in\GL_n(\Zp)$
- hard to use
history
- 1993 : Gross-Keating : introduced $GK(B)$ when $B$ is of size $3$ in the study of arithmetic intersection number related to three modular polynomials
- 2015 : Ikeda-Katsurada : defined $GK(B)$ for $B$ half-integral of arbitary size over a finite extension of $\Qp$
- 2016 : Ikeda-Katsurada : showed that the Siegel series of $B$ (this is something appearing as a local factor of Fourier coefficient of Siegel-Eisenstein series) is determined by $GK(B)$
- 2017 : Cho-Ikeda-Katsurada-Yamauchi : many computer-friendly (not human-friendly) formulas (and I recently wrote computer program using Mathematica)
binary quadratic forms and class number relations
- $Q=Ax^2+Bxy+Cy^2$ : positive definite binary quadratic form over $\Z$, write $Q=[A,B,C]$
- discriminant of $Q$ : $B^2-4AC<0$
- for positive integer $d$, define
- $\mathcal{Q}_d=\{Q:B^2-4AC=-d\}$
- $\mathcal{Q}_{d}^{pr}=\{Q\in \mathcal{Q}_d:\text{primitive}\}$. $Q$ is primitive if $A,B,C$ is coprime
- $\Gamma=PSL_2(\mathbb{Z})$ acts on $\mathcal{Q}_d$ (and $\mathcal{Q}_{d}^{pr}$) : $Q\mapsto Q'$ by $Q'(x,y)=Q(ax+by,cx+dy)$, in matrix form,
$$ \left( \begin{array}{cc} A & \frac{B}{2} \\ \frac{B}{2} & C \\ \end{array} \right) \mapsto \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)^t\left( \begin{array}{cc} A & \frac{B}{2} \\ \frac{B}{2} & C \\ \end{array} \right)\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) $$
- for each $Q$, let $w_{Q}$ be the size of stabilizers
- $w_Q=2$ if $Q\sim [a,0,a]$
- $w_Q=3$ if $Q\sim [a,a,a]$
- $w_Q=1$ otherwiser
- def (class number and Hurwitz-Kronecer class number)
For pos. int. $d>0$, define $$h_{d}^{pr}:=\sum_{Q\in \mathcal{Q}_d^{pr}/\Gamma} 1$$
$$h_d:=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}$$
- 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 $d=23$, they are the same : $\{[1,1,6], [2,-1,3], [2,1,3]\}$, $h_{d}=h_{d}^{pr} = 3$
class polynomials
- def (j-invariant)
- $
j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+21493760q^2+\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= 1+240q+2160q^2+\cdots,\quad \sigma_3(n)=\sum_{d|n}d^3$
- $\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots$
- thm
Let $Q=[A,B,C]$ be primitive of discriminant $-d$, and $\tau_Q = \frac{-B+\sqrt{B^2-4AC}}{2A}\in \mathbb{H}$. Then $j(\tau_Q)$ is an algebraic integer with minimal polynomial $$ H_d(x) : = \prod_{Q\in \mathcal{Q}_{d}^{\rm{pr}}/\Gamma}(x-j(\tau_Q))\in \Z[x] $$ In particular, $h_{d}^{\rm{pr}}=1$, then $j(\tau_Q)\in \mathbb{Z}$.
- $\Delta =-163$; $h_{163}^{pr}=1$
$$j(\frac {-1+\sqrt{-163}} {2})=-640320^3$$
- $\Delta =-23$; $h_{23}^{pr}=3$
$$ H_{23}(x) = x^3+3491750 x^2-5151296875 x+12771880859375 $$
modular polynomials
- Let $m$ be a positive integer
- there exists $\phi_m(x,y)\in{\mathbb{Z}}[x,y]$ such that
$$\prod_{ad=m,1\leq b \leq d}(x-j(\frac{a\tau+b}{d}))=\phi_m(x,j(\tau))$$
- as a polynomial in $x$, $\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d$
- examples
- $m=1$, $\phi_1(x,y)=x-y$
- $m=2$
$$ \phi_2(x,y)=x^3+y^3-x^2 y^2+1488 (x^2 y + x y^2)-162000 (x^2+y^2) +40773375 x y+8748000000 (x + y)-157464000000000 $$
- $m=3$
$$ \begin{aligned} \phi_3(x,y) =x^4+x^3 \left(-y^3+2232 y^2-1069956 y+36864000\right)+\dots \end{aligned} $$
- $m=4$
$$ \phi_4(x,y) = x^7+x^6 \left(-y^4+2976 y^3-2533680 y^2+561444610 y-8507430000\right)+\dots $$
- we are interested in $F_m(x):=\phi_m(x,x)\in \Z[x]$ :
$$ F_1(x)=0 $$ $$ F_2(x) = -(x-1728)(x+3375)^2(x-8000) = -H_{4}(d)H_{7}(x)^2H_{8}(x) $$ $$ F_3(x) = -x(x-8000)^2 (x+32768)^2(x-54000) = - H_3(x)H_{8}(x)^2H_{11}(x)^2H_{12}(x) $$
- if $m$ is not a perfect square, $F_m(x)$ is non-zero.
- Hurwitz calculated its degree :
$$\deg F_m(x)= \sum_{d|m}\max(d,m/d)$$
- Kronecker gave its explicit factorization in terms of class polynomials:
$$ F_m(x) =\pm \prod_{t\in \Z,t^2 \leq 4m}\mathcal{H}_{4m − t^2}(x) $$ where $$ \mathcal{H}_d(x) = \prod_{Q\in \Gamma\backslash \mathcal{Q}_d}(x-j(\tau_Q))^{1/w_{Q}} $$
- it can be written as a product of class polynomials $H_d(x)$'s with known exponents.
- thm (Kronecker-Hurwitz class number relation)
If $m$ is not a perfect square, 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 discriminants 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 actually computed this for pairs $\phi_{m_1}$ and $\phi_{m_2}$
arithmetic intersection number
- thm (Gross-Keating)
Let $m_1,m_2,m_3$ be positive integers.
- The cardinality of the quotient ring $A=\Z[X,Y]/(\phi_{m_1},\phi_{m_2},\phi_{m_3})$ is finite if and only if there is no positive definite integral binary quadratic form $a x^2+bxy+cy^2$ which represents the three integers $m_1,m_2,m_3$.
- Assume that $m_1,m_2,m_3$ satisfy this condition. Let $\log \#A=\sum_{p}n(p)\log p$. Then $n(p)=0$ for $p>4m_1m_2m_3$. For $p\leq 4m_1m_2m_3$,
$$ n(p) = \frac{1}{2}\sum_{Q}\left(\prod_{l\mid 4\det Q,\, l\neq p} \beta_l(Q) \right)\cdot \alpha_p(Q). $$
- Here the sum is over all positive definite non-degenerate half-integral matrices $Q$ 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 $\beta_p(Q)$ are given explicitly in terms of $GK(Q)=(a_1,a_2,a_3)$ ($Q$ as a matrix over $\Qp$). For example,
If $a_1\not\equiv a_2 \pmod 2$, $$ \alpha_p(Q) = \sum_{i=0}^{a_1-1} (i+1) (a_1+a_2+a_3-3 i)p^i +\sum _{i=a_1}^{(a_1+a_2-1)/2} (a_1+1) (2a_1+a_2+a_3-4i)p^i. $$ $$ \beta_p(Q) = \sum _{i=0}^{a_1-1} 2(i+1)p^i +\sum _{i=a_1}^{(a_1+a_2-2)/2} 2(a_1+1)p^i. $$
- Let $S=\mathrm{Spec}\, \Z[X,Y]$ and $T_m$ be the divisor on $S$ corresponding to $\phi_m$. $\intmult$ is called the arithmetic intersection number.
representation of integers by quadratic forms
- Let $Q$ be a positive definite integral quadratic form in $n$ variables, i.e. $Q(X) = X^t A_{Q} X$ for some positive definite half-integral symmetric square matrix $A_{Q}$
- $r(Q, m)$ : 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 integer $N$ such that $N(2A_Q)^{-1}$ is twice of a half-integral matrix
- for example, $Q=x^2$, $\det Q = 2$, $N=4$; $Q=4x^2+6y^2$, $\det Q = 96$, $N=48$
- thm (see Theta function of a quadratic form)
For simplicity assume that $Q$ has even number of variables. (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. $\Gamma_0(N)$
- vector 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 coefficient of $E_Q(\tau)$ + Fourier coefficient of $C_Q(\tau)$ (i.e. dominant term + error term)
Siegel-Weil formula
- key message : do not study a single form. consider all forms in its genus
- def (genus of an integral quadratic form)
For $Q$, ${\rm gen}(Q)$ is the set of $\Z$-equivalence classes of quadratic forms that are $\Z_p$-equivalent to $Q$ at all $p$ (including $p=\infty$)
When $Q$ is positive definite, ${\rm gen}(Q)$ is a finite set.
- example
$f_1(x,y) =x^2+82y^2$ and $f_2(x,y) =2x^2+41y^2$ are $\Zp$-equivalent for all $p$ , but not $\Z$-equivalent
- thm (Siegel)
Let $Q$ be a positive definite quadratic form $Q$ over $\Z$. To each $Q' \in {\rm gen}(Q)$, assign the 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 is the Eisenstein part $E_{Q}$ of $\theta_Q$ $$\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)$$
- weighed average of representation number, i.e. the Fourier coefficient of $E_Q$ is
$$ \sum_{Q'\in {\rm gen}(Q)}w(Q')r(Q', m)=\prod_{p:\text{primes}}\alpha_{p}(Q,m) = \alpha_{\infty}(Q,m)\alpha_{2}(Q,m)\alpha_{3}(Q,m)\dots $$ where $\alpha_{p}(Q,m)$ is the local density at $p$ (will be defined soon).
- remark
- Let $A$ and $B$ be half-integral symmetric square matrices over $\Z$ of size $m$ and $n$, $m\geq n\geq 1$
- define $r(A,B)$ to be the number of $m \times n$ integer matrices $X$ such that $X^t A X = B$.
- 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 $$ N_{p^{\ell}}(A,B) = \#\{X\in M_{m\times n}(\Zp/p^{\ell}\Zp)\, | X^{t}AX = B \pmod{p^{\ell}\calh_n(\Zp)}\} $$
- it is very difficult to compute $\alpha_{p}(A,B)$ in general
- there is an important special case we know more
- thm (?Kitaoka)
Let $B\in \matn$. There exists a polynomial $f_p(B;X)\in \Z[X]$ such that for $2k\geq n$, $$ f_p(B;p^{-k}) = \alpha_{p}(H_{k},B) $$ 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)
the polynomial $f_p(B;X)$ (more precisely, $f_p(B;p^{-s}),\, s\in \C$) is called the Siegel series of $B$
- remark
- the Siegel series appear as the $p$-factors of the Fourier coefficients of the Siegel-Eisenstein series (for the full modular group $\operatorname{Sp}(n,\Z)$, or weighted average for even unimodular lattices)
- thm (Ikeda-Katsurada 2016)
The Siegel series of $B$ is determined by $GK(B)=(a_1,\dots, a_n)$ and 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-integral positive semi-definite symmetric $g\times g$ 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 coefficients 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 symmetric 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-degenerate $T\in\Xgsemi$, $$ \fc T\Egk= \dfrac{2^{\lfloor \frac{g+1}{2} \rfloor} \prod_{p}F_p(T,p^{k-g-1})} {\zeta(1-k)\prod_{i=1}^{\lfloor g/2\rfloor}\zeta(1-2k+2i)} \cdot\begin{cases} L_{D_T}(1-k+g/2)&\text{$g$ even},\\ 1&\text{$g$ odd} \end{cases} $$ 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
- 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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