J-불변량과 모듈라 다항식

개요

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

\(n=1\)

\[ \Phi_1(x,y)=x-y \]

\(n=2\)

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

\(n=3\)

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

\(n=4\)

\[ \Phi_4(x,y)=x^6+y^6+\dots \]

class number relation

\(m>0\) : int
\(\exists\) \(\phi_m(x,y)\in{\mathbb{Z}}[x,y]\) such that

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

\(\phi_m(j(m\tau),j(\tau))=0\)
\(\phi_{m}=\prod_{n^2|m}\Phi_{m/n^2}\)
as a poly. in \(x\), \(\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d\)

examples

\(m=1\), \(\phi_1(x,y)=\Phi_1(x,y)\)
\(m=2\), \(\phi_2(x,y)=\Phi_2(x,y)\)
\(m=3\), \(\phi_3(x,y)=\Phi_3(x,y)\)
\(m=4\), \(\phi_4(x,y) = \Phi_1(x,y)\Phi_4(x,y) = x^7+\dots\)
interested in \(F_m(x):=\phi_m(x,x)\in \Z[x]\) :

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

\(F_m(x)\neq 0\) 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 \[ \mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}} \]

Let \(h_d\) be the Hurwitz-Kronecer class number (후르비츠-크로네커 유수)

\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}

thm (class number relation ver. 1)

For \(m\) is not a perfect sq., define \[ G(m)= \sum_{t\in \Z,t^2 \leq 4m}h_{4m − t^2} \] Then \[ G(m)=\sum_{d|m}\max(d,m/d) \]

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 (class number relation ver. 2)

Assume that \(m=m_1m_2\) is not a perfect square. Let \(A=\C[X,Y]/\langle \phi_{m_1},\phi_{m_2}\rangle\). Then \[ |A|=\sum_{n|\gcd(m_1,m_2)}nG(m/n^2) \]

테이블

\begin{array}{c|cccccccccc} \left\{m_1,m_2\right\} & \{2,3\} & \{2,4\} & \{2,5\} & \{2,6\} & \{2,7\} & \{2,9\} & \{3,4\} & \{3,5\} & \{3,6\} & \{3,7\} \\ \hline \sum_{n|\gcd(m_1,m_2)}nG(m_1m_2/n^2) & 18 & 32 & 30 & 56 & 42 & 66 & 44 & 40 & 78 & 56 \\ \end{array}

J-불변량과 모듈라 다항식

목차

개요

예

class number relation

테이블

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