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

$ \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 invariant

• Let $p\in \Z_{\geq 0}$ be a prime, $\Qp$, and $\Zp$ its ring of integers
• For $a\in \Qp^\times$, $\ord(a)=n$ if $a\in p^n \Zp^\times$
• $B=(b_{ij})\in \sym{\Qp}$ is half-integral if $2b_{ij}\in \Zp$, and $b_{ii}\in \Zp$ for any $i,j$
• $\matn$ : set of non-degenerate half-integral symmetric matrix of size $n$
• $B,B'\in\matn$, $B\sim_{R} B'$ if there exists $U\in\GL_n(\Zp)$ such that $B' = U^{t}BU$

definition

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 $\GK(B)=(a_1, \dots, a_n)$ 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*}

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 $B$ matrix of arbitary size over a finite extension of $\Zp$
• 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 $GK(B)$

binary quadratic forms and class number relations

• $Q=[A,B,C]=Ax^2+Bxy+Cy^2$ : positive definite binary quadratic form over $\Z$
• $Q$ is primitive if $A,B,C$ is coprime
• discriminant of Q : $\Delta=B^2-4AC$
• $\mathcal{Q}_d=\{Q:B^2-4AC=-d\}$
• $\mathcal{Q}_{d;prim}=\{Q\in \mathcal{Q}_d:\text{primitive}\}$
• $\Gamma=PSL(2,\mathbb{Z})$ acts on $\mathcal{Q}_d$ : $Q\mapsto Q'$ 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 $$h_{d;prim}:=\sum_{Q\in \mathcal{Q}_{d;prim}/\Gamma} 1$$

$$h_d:=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}$$

• we set $h_0=-1/12$

\begin{array}{cccccccccccccc} d & 0 & 3 & 4 & 7 & 8 & 11 & 12 & 15 & 16 & 19 & 20 & 23 & 24 \\ h_{d;prim} & . & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 3 & 2 \\ h_d & -\frac{1}{12} & \frac{1}{3} & \frac{1}{2} & 1 & 1 & 1 & \frac{4}{3} & 2 & \frac{3}{2} & 1 & 2 & 3 & 2 \\ \end{array}

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 $$ H_d(x) = \prod_{Q\in \mathcal{Q}_{d;\rm{prim}}/\Gamma}(x-j(\tau_Q))\in \Z[x] $$ In particular, $h_{d;prim}=1$, then $j(\tau_Q)\in \mathbb{Z}$.

• $\Delta =-163$; $h_{163}=1$

$$j(\frac {-1+\sqrt{-163}} {2})=-640320^3$$

• $\Delta =-23$; $h_{23}=3$

$$ x^2+xy+6y^2, 2x^2-xy+3y^2, 2x^2+xy+3y^2 $$ $$ j\left(\frac{1}{2} \left(-1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(-1+i \sqrt{23}\right)\right)$$ $$ H_{23}(x) = x^3+3491750 x^2-5151296875 x+12771880859375 $$

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 $F_m(x):=\phi_m(x,x)\in \Z[x]$ :

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

• 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!

arithmetic intersection number

• Let $m_1,m_2,m_3$ be positive integers

thm (Gross-Keating)

1. 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 binary quadratic form $a x^2+bxy+cy^2$ with $a,b,c\in \Z$ which represents the three integers $m_1,m_2,m_3$.
2. 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 $p\leq 4m_1m_2m_3$,

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

• Here $\Delta = 4\det Q\in \Z$ and 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 $GK(Q)=(a_1,a_2,a_3)$, in which $Q$ is regarded as an element $\mat{3}{p}$. 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 function of a quadratic form)

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

thm

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. $\Gamma_0(N)$

• what is $N$?
• what is $\det Q$?

• 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 $C_Q(\tau)$ (dominant term + error term)

Siegel-Weil formula

• 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 $\Z$. For each $Q' \in {\rm gen}(Q)$, assign the weight $$ w(Q') = \frac{1}{|{\rm Aut}(Q')|}\,\cdot\,\left(\sum_{Q'\in {\rm gen}(Q)}\frac{1}{|{\rm Aut}(Q')|}\right)^{-1} $$ so that $\sum_{Q'} w(Q')=1$.

1. weighted average of theta functions is the Eisenstein part $E_{Q}$ of $\theta_Q$ $$\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)$$
2. 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 $n$, respectively.
• define $r(A,B)$ to be the number of $m \times n$ integral matrices $X$ such that $X^t A X = B$.
• Siegel's theorem holds for $r(A,B)$
• modular form -> 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 $$ N_{p^{\ell}}(A,B) = \#\{X\in M_{m\times n}(\Zp/p^{\ell}\Zp)\, | X^{t}AX = B \pmod{p^{\ell}\calh_n(\Zp)}\} $$

• computing $\alpha_{p}(A,B)$ is very difficult

thm (?Kitaoka)

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

def (Siegel series)

The function $b_p(B;s) =f_p(B;p^{-s})$ is called the Siegel series.

remark

• the Siegel series appear as the $p$-factors of the Fourier coefficients of the Siegel-Eisenstein series (for the full modular group $\operatorname{Sp}(n,\Z)$)

thm (Ikeda-Katsurada 2016)

The Siegel series $b_p(B;s)$ 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 $p\mid2\det(2T)$.

related items

• 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
• 틀:수학노트
• 틀:수학노트
• 틀:수학노트
• 틀:수학노트

computational resource

• https://drive.google.com/file/d/1kj7C_5xBW8nGMQiOOAc31nXTfAeP8ykG/view