Talk on Gross-Keating invariants

overview

definition 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 invariant 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 invariant

$p\in \Z_{> 0}$ : 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. matrices

def (Gross-Keating invariant)

$B=(b_{ij})\in\matn$

$S(B)$ : 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*}

$S(\{B\}):=\bigcup_{U\in\GL_n(\Zp)} S(U^{t}BU)$

Gross-Keating invariant $\GK(B)=(a_1, \dots, a_n)\in\Zn$ 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 $B$ (a local factor of Fourier coef of Siegel-Eisenstein series) is determined by $GK(B)$
2017 : Cho-Ikeda-Katsurada-Yamauchi : many 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-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

$Q=[A,B,C]$ : 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

$$\prod_{ad=m,0\leq b \leq d-1}(x-j(\frac{a\tau+b}{d}))=\phi_m(x,j(\tau))$$

as a poly. in $x$, $\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d$

examples

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

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

$m=3$

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

$m=4$

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

interested in $F_m(x):=\phi_m(x,x)\in \Z[x]$ :

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

$F_m(x)\neq 0$ 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. non-deg. 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 $\beta_p(Q)$ are given explicitly 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 corresponding to $\phi_m$ on $S=\mathrm{Spec}\, \Z[X,Y]$

repn of integers by quad. forms

$Q$ : a pos. def. quad. form $/\Z$ in $n$ variables, 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 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. $\Gamma_0(N)$

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 : $$\sum_{Q'\in {\rm gen}(Q)}w(Q')\theta_{Q'}(\tau)=E_{Q}(\tau)$$
weighed 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 $p\mid2\det(2T)$.

related items

computational resource

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