Talk on Gross-Keating invariants

overview

The Gross-Keating invariant of a quadratic form over p-adic integers is a relatively recent but fundamental concept in the study of quadratic forms
classical modular polynomials for the j-invariant
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(\frko)^{\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, $F=\Qp$, and $\frko=\Zp$ its ring of integers.
For $a\in F^\times$, we write $\ord(a)=n$ if $a\in p^n \frko^\times$, and call it the valuation of $a$, and set $\ord(0)=\infty$.
For a subring $R$ of $F$ containing $\frko$, we denote the set of symmetric square matrices of degree $n$ with entries in $R$ by $\sym{R}$.
We say $B=(b_{ij})\in \sym{F}$ is half-integral if $2b_{ij}\in \frko$, and $b_{ii}\in \frko$ for any $i,j$ and denote the set of non-degenerate half-integral symmetric matrix of degree $n$ by $\matn$.
For $B\in \matn$, we write $\deg(B)=n$.
When there exists $U\in\GL_n(R)$ such that $B' = U^{t}BU$ for $B,B'\in\matn$, we say they are $R$-equivalent and write $B\sim_{R} B'$.

definition

Let $B=(b_{ij})\in\calh_n(\frko)^{\rm nd}$. 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 $S(\{B\}):=\bigcup_{U\in\GL_n(\frko)} S(U^{t}BU)$. 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*}

j-invariant

$q=e^{2\pi i\tau},\tau\in \mathbb{H}$
j-invariant

$

j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+21493760q^2+\cdots $ 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$

$\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots$

Eisenstein series

$E_{2k}(\tau)=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)= 1-\frac {4k}{B_{2k}}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)$

$$ \begin{aligned} E_4(\tau)&=1+ 240\sum_{n=1}^\infty \sigma_3(n) q^{n}&=&1 + 240 q + 2160 q^2 + 6720 q^3 + 17520 q^4 + 30240 q^5+\cdots \\ E_6(\tau)&=1- 504\sum_{n=1}^\infty \sigma_5(n) q^{n}&=&1 - 504 q - 16632 q^2 - 122976 q^3 - 532728 q^4 - 1575504 q^5+\cdots \\ E_8(\tau)&=1+ 480\sum_{n=1}^\infty \sigma_7(n) q^{n}&=&1+480 q+61920 q^2+1050240 q^3+7926240 q^4+37500480 q^5+\cdots \\ \end{aligned} $$

integral binary quadratic forms

$Q=ax^2+bxy+cy^2$ is primitive if $a,b,c$ is coprime
판별식 $\Delta=b^2-4ac$
$f(x,y)=g(ax+by,cx+dy)$ 인 정수 $a,b,c,d,\, ad-bc= 1$가 존재하면, $f\sim g$ 이라 함

$\mathcal{Q}_d$는 $-d=b^2-4ac$를 만족하는 정수계수 이변수 이차형식(binary integral quadratic forms) $Q=[a,b,c]=ax^2+bxy+cy^2$의 집합
$\Gamma=PSL(2,\mathbb{Z})$은 $\mathcal{Q}_d$에 작용
각각의 $Q$에 대하여, 자기동형군 $\Gamma_{Q}$을 생각, $w_{Q}=|\Gamma_{Q}|$
- $w_Q=2$ if $Q\sim [a,0,a]$
- $w_Q=3$ if $Q\sim [a,a,a]$
- 다른 경우 $w_Q=1$
후르비츠-크로네커 수를 다음과 같이 정의

$H(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma} \frac{1}{w_Q}$

i.e. the number of equivalence classes of positive definite binary quadratic forms over Z with determinant D, except that the forms equivalent to

$[a,0,a]$ [resp. $[a,a,a]$ are counted with multiplicity 1/2 [resp. 1/3]

\begin{array}{c|ccccccccccccccccccccccccc} d & 0 & 3 & 4 & 7 & 8 & 11 & 12 & 15 & 16 & 19 & 20 & 23 & 24 & 27 & 28 & 31 & 32 & 35 & 36 & 39 & 40 & 43 & 44 & 47 & 48 \\ \hline 12 H(d) & -1 & 4 & 6 & 12 & 12 & 12 & 16 & 24 & 18 & 12 & 24 & 36 & 24 & 16 & 24 & 36 & 36 & 24 & 30 & 48 & 24 & 12 & 48 & 60 & 40 \\ \end{array}

class polynomial

thm

$ H_d(x) = \prod_{Q\in \Gamma\backslash \mathcal{Q}_{d;\rm{prim}}}(x-j(\tau_Q))\in \Z[x] $

Thus, $j(\tau_Q)$ is an algebraic integer of degree $h_d$ and $h_d=1$, then $j(E)\in \mathbb{Z}$

$\Delta =-163$; $h_{163}=1$

$$j(\frac {-1+\sqrt{-163}} {2})=-262537412640768000=-640320^3$$

$\Delta =-23$

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

we slightly modify $H_d$ as follows :

$$ \mathcal{H}_d(x) = \prod_{Q\in \Gamma\backslash \mathcal{Q}_d}(x-j(\tau_Q))^{1/w_{Q}} $$

modular polynomials

Let $m$ be a positive integer
there exsits a polynomial $\phi_m(x,y)\in{\mathbb{Z}}[x,y]$ 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 $x$, $\deg \phi(n)=\sigma_1(m)=\sum_{d|m}d$

examples

$m=1$, $\phi_1(x,y)=x-y$
$m=2$

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

$m=3$

$$ \begin{aligned} \phi_3(x,y) =x^4+x^3 \left(-y^3+2232 y^2-1069956 y+36864000\right)+\dots \end{aligned} $$

$m=4$

$$ \phi_4(x,y) = x^7+x^6 \left(-y^4+2976 y^3-2533680 y^2+561444610 y-8507430000\right)+\dots $$

prop (Hurwitz)

$\deg \phi_m(x,x)= \sum_{d|m}\max(d,m/d)$

prop (Kronecker)

if $m$ is not a perfect square, $\phi_m(x,x)\in \Z[x]$ is non-zero
the Kronecker-Hurwitz relation

$$ \sum_{d|m}\max(d,m/d) = \sum_{t\in \Z}H(4m − t^2),\, (t^2 \leq 4m) $$

representation of a number by a form

theta function

theta function of a quadratic form = Eisenstein series + cusp form

Siegel-Weil formula

thm

For a positive definite even unimodular lattice $L$, $$\left( \sum_{M\in {\rm gen}(L)}\frac{\Theta_M^{(g)}(Z)}{|{\rm Aut}(M)|}\right)\,\cdot\, \left(\sum_{M\in {\rm gen}(L)}\frac{1}{|{\rm Aut}(M)|}\right)^{-1}= E^{(g)}_{k}(Z),$$

Moreover, the Fourier coefficients $a_{E}(N)$ of $E$ can be expressed as an infinite product of local densities $$ a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp} $$

representation of a form by a form

Let $A$ and $B$ be symmetric square matrices with entries in $\Z$ of size $m$ and $n$, respectively. Define $r(A,B)$ to be the number of $m \times n$ integral matrices $X$ such that $X^t A X = B$.

def

For an $m$-dimensional lattice $\Lambda$, define $r(\Lambda,B):= r(A,B)$, where $A$ is a Gram matrix of $\Lambda$.

If $M$ is positive-definite, then $r(M,\cdot)$ is finite.
$r(A,A)$ gives the size of the automorphism group of $M$.

Local density and Siegel series

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 semi-integral positive semi-definite symmetric $g\times g$ matrices. A matrix is \textit{semi-integral} if it has entries in $\frac{1}{2}\mathbb{Z}$ with integral diagonal. }

\frame{\frametitle{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 summation extends over all classes of coprime symmetric pairs, i. e. over all inequivalent bottom rows of elements of $\Gamma_g$ with respect to left multiplications by unimodular integer matrices of degree $g$. In other words, the sum is over a full set of representatives for the cosets $\GLnZ\backslash \Gamma_{g}$)

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

\frame{\frametitle{Formula for the Fourier coefficients of Eisenstein series} %\textbf{Q. state the condition on $k$ and $g$ for the formula} Consider the Fourier expansion of $\Egk(Z)$ : $$ \Egk(Z)=\sum_{T\in\Xgsemi}\fc T{\Egk}\,\e(\ip TZ). $$

thm (Kitaoka?)

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 $p\mid2\det(2T)$.

arithmetic intersection number

formula of \cite{MR1213101} for the arithmetic intersection number of three modular correspondences from a computational perspective.
As mentioned in the Introduction, this is the original context in which the Gross-Keating invariants have been introduced for ternary quadratic forms over $\Zp$.
Let us denote the set of non-degenerate half-integral matrices with entries in $\Z$ by $\Zmat{n}$.
We can regard $Q\in \Zmat{n}$ as an element of $\mat{n}{p}$ for any prime $p$.

For $m\in \Z_{\geq 1}$, let $\phi_m(X,Y)\in \Z[X,Y]$ be the classical modular polynomial
Let $m_1,m_2,m_3\in \Z_{\geq 1}$.
Gross and Keating showed that the cardinality of the quotient ring $\Z[X,Y]/(\phi_{m_1},\phi_{m_2},\phi_{m_3})$ 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 $m_1,m_2,m_3$. Assume that $m_1,m_2,m_3$ satisfy this condition.
Let $S=\mathrm{Spec}\, \Z[X,Y]$ and $T_m$ be the divisor on $S$ corresponding to $\phi_m$. We define the arithmetic intersection number as follows :

\begin{equation}\label{eqn:TTT} \begin{aligned} \intmult : & = \log \# \Z[X,Y]/(\phi_{m_1},\phi_{m_2},\phi_{m_3}) \\ & = \sum_{p}n(p)\log p, \end{aligned} \end{equation} with $n(p)=0$ for $p>4m_1m_2m_3$. Furthermore, Gross and Keating found an explicit formula for $n(p)$.

thm

Let $p$ be a prime. We have $$ n(p) = \frac{1}{2}\sum_{Q}\left(\prod_{l\mid \Delta,\, l\neq p} \beta_l(Q) \right)\cdot \alpha_p(Q), $$ with $\Delta = 4\det Q\in \Z$. The sum is over all positive definite matrices $Q\in \Zmat{3}$ with diagonal $(m_1,m_2,m_3)$ which are isotropic over $\Q_{l}$ for all $l\neq p$. Such $Q$ is anisotropic over $\Qp$ and $p$ divides $\Delta$. The quantities $\alpha_p(Q)$ and $\beta_p(Q)$ are given as follows : Let $H = (a_1, a_2, a_3; \vep_1, \vep_2, \vep_3)$ be a naive EGK datum of $Q$ at regarded as elements of $\mat{3}{p}$, as in (\ref{eqn:NEGKodd}) and (\ref{eqn:NEGKeven}).

When $a_1\equiv a_2 \pmod 2$ and $a_2<a_3$, we further define $\epsilon$ to be $\vep_2$.

For example,

If $a_1\not\equiv a_2 \pmod 2$, then $\alpha_p(Q)$ is equal to $$ \sum_{i=0}^{a_1-1} (i+1) (a_1+a_2+a_3-3 i)p^i +\sum _{i=a_1}^{(a_1+a_2-1)/2} (a_1+1) (2a_1+a_2+a_3-4i)p^i. $$

If $a_1\not\equiv a_2 \pmod 2$, then $\beta_p(Q)$ is equal to $$ \sum _{i=0}^{a_1-1} 2(i+1)p^i +\sum _{i=a_1}^{(a_1+a_2-2)/2} 2(a_1+1)p^i. $$

related items

computational resource

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