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
• the classical modular polynomials for the j-invariant
• such as the representation of integers by quadratic forms
• The rest of the talk will be devoted to a computer program that computes the Gross-Keating invariant of a quadratic form over Zp, and other related quantities

$ \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 two square matrices $X$ and $Y$ with entries in $F$, we denote the matrix $\begin{pmatrix} X & O \\ O & Y\end{pmatrix}$ by $X\bot Y$. We denote the diagonal matrix $(b_1)\bot \dots \bot (b_n)$ by $\diag(b_1, \dots, b_n)$. 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

• \(q=e^{2\pi i\tau},\tau\in \mathbb{H}\)라 두자
• 타원 모듈라 j-함수는 다음과 같이 정의된다

\[ j(\tau)= {E_ 4(\tau)^3\over \Delta(\tau)}=q^{-1}+744+196884q+21493760q^2+\cdots \] 여기서 \[ E_ 4(\tau)=1+240\sum_{n>0}\sigma_3(n)q^n= 1+240q+2160q^2+\cdots,\quad \sigma_3(n)=\sum_{d|n}d^3\]는 아이젠슈타인 급수(Eisenstein series), \[\Delta(\tau)= q\prod_{n>0}(1-q^n)^{24}= q-24q+252q^2+\cdots\] 는 판별식 함수

singular moduli

• quadratic imaginary number 에서의 값

thm

If $E$ has complex multiplication, then $j(E)$ is an algebraic integer of degree $h_K$. Especially if $h_K=1$, then $j(E)\in \mathbb{Z}$

• 예 :

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

• 타원 모듈라 j-함수의 singular moduli 참조
• 판별식이 -23인 세 이차형식 (숫자 23과 다항식 x³-x+1 참조)

$$ x^2+x+6,2 x^2-x+3,2 x^2+x+3 $$ 의 상반평면에서의 해를 구하여, 다음의 값을 생각하자 $$ j\left(\frac{1}{2} \left(-1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(1+i \sqrt{23}\right)\right),j\left(\frac{1}{4} \left(-1+i \sqrt{23}\right)\right)$$

• 이들은 대수적 정수이며, 다음 다항식의 해가 된다

$$ x^3+3491750 x^2-5151296875 x+12771880859375 $$

integral binary quadratic forms

• 판별식\[\Delta=b^2-4ac\]
• primitive 이차형식은 \(a,b,c\) 가 서로소인 이차형식 \(ax^2+bxy+cy^2\)으로 정의됨

정수계수 이변수 이차형식의 동치류

• 다음 두 변환에 의해 같아지는 이차형식은 모두 같은 동치류에 있다고 정의

\[x \to x+y, y \to y\] \[x \to x, y \to x+y\]

• 이러한 변환을 행렬로 표현하면 각각 다음과 같으며, 이는 모듈라 군(modular group)을 생성함

\[T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} , R=\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \] $$ S=\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right) $$로 두면, $S=T^{-1}RT^{-1}$

• 즉 \(f(x,y)=g(ax+by,cx+dy)\) 인 정수 $a,b,c,d,\, ad-bc= 1$가 존재하면, \(f\sim g\) 이라 함

modular polynomials

• 타원 모듈라 j-함수 (elliptic modular function, j-invariant)
• $\Phi_n\bigl(j(n\tau),j(\tau)\bigr)=0$를 만족하는 기약다항식 $\Phi_n(x,y)\in{\mathbb{

Z}}[x,y]$이 존재하며, 이 때 차수는 $x,y$ 각각에 대하여 $\psi(n)=n\prod_{p|n}(1+1/p)$로 주어진다

예

• $n=2$

$$ \Phi_2(x,y)=x^3+y^3-x^2 y^2+1488 (x^2 y + x y^2)-162000 (x^2+y^2) +40773375 x y+8748000000 (x + y)-157464000000000 $$

• $n=3$

$$ \begin{aligned} \Phi_3(x,y) &=x^4+y^4-x^3 y^3+36864000 \left(x^3+y^3\right)-1069956 \left(x^3 y+x y^3\right)+2587918086 x^2 y^2 \\ &+452984832000000 \left(x^2+y^2\right)+8900222976000 \left(x^2 y+x y^2\right)+2232 \left(x^3 y^2+x^2 y^3\right) \\ &-770845966336000000 x y+1855425871872000000000 (x+y) \end{aligned} $$

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 $$ a_{E}(N)=\prod_{p:\text{primes}}\beta_{L,p}(N) \label{lp} $$

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

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 $M$.

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

In this section, we consider the 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; see \cite{MR1213101} and the references therein. 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 (Proposition 3.22]{MR1213101}\label{thm

GKformula})

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

• 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