Talk on Gross-Keating invariants
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
$$ x^3+3491750 x^2-5151296875 x+12771880859375 $$
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
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
- $n=2$
$$ \Phi_2(x,y)=x^3+y^3-x^2 y^2+1488 (x^2 y + x y^2)-162000 (x^2+y^2) +40773375 x y+8748000000 (x + y)-157464000000000 $$
- $n=3$
$$ \begin{aligned} \Phi_3(x,y) &=x^4+y^4-x^3 y^3+36864000 \left(x^3+y^3\right)-1069956 \left(x^3 y+x y^3\right)+2587918086 x^2 y^2 \\ &+452984832000000 \left(x^2+y^2\right)+8900222976000 \left(x^2 y+x y^2\right)+2232 \left(x^3 y^2+x^2 y^3\right) \\ &-770845966336000000 x y+1855425871872000000000 (x+y) \end{aligned} $$