"J-불변량과 모듈라 다항식"의 두 판 사이의 차이

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==개요==
 
==개요==
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)]]
 
* [[타원 모듈라 j-함수 (elliptic modular function, j-invariant)]]
−
* $\Phi_n\bigl(j(n\tau),j(\tau)\bigr)=0$를 만족하는 기약다항식 $\Phi_n(x,y)\in{\mathbb{
+
* <math>\Phi_n\bigl(j(n\tau),j(\tau)\bigr)=0</math>를 만족하는 기약다항식 <math>\Phi_n(x,y)\in{\mathbb{
−
Z}}[x,y]$이 존재하며, 이 때 차수는 $x,y$ 각각에 대하여 $\psi(n)=n\prod_{p|n}(1+1/p)$로 주어진다
+
Z}}[x,y]</math>이 존재하며, 이 때 차수는 <math>x,y</math> 각각에 대하여 <math>\psi(n)=n\prod_{p|n}(1+1/p)</math>로 주어진다
  
  
 
==예==
 
==예==
−
* $n=2$인 경우
+
* <math>n=1</math>
−
$$
+
:<math>
 +
\Phi_1(x,y)=x-y
 +
</math>
 +
* <math>n=2</math>
 +
:<math>
 
\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
 
\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
−
$$
+
</math>
−
* $n=3$인 경우
+
* <math>n=3</math>
−
$$
+
:<math>
 
\begin{aligned}
 
\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 \\
 
\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 \\
17번째 줄: 21번째 줄:
 
&-770845966336000000 x y+1855425871872000000000 (x+y)
 
&-770845966336000000 x y+1855425871872000000000 (x+y)
 
\end{aligned}
 
\end{aligned}
−
$$
+
</math>
 +
* <math>n=4</math>
 +
:<math>
 +
\Phi_4(x,y)=x^6+y^6+\dots
 +
</math>
 +
 
 +
 
 +
==class number relation==
 +
* <math>m>0</math> : int
 +
* <math>\exists</math> <math>\phi_m(x,y)\in{\mathbb{Z}}[x,y]</math> such that
 +
:<math>\prod_{ad=m,a,d>0,0\leq b \leq d-1}(x-j(\frac{a\tau+b}{d}))=\phi_m(x,j(\tau))</math>
 +
* <math>\phi_m(j(m\tau),j(\tau))=0</math>
 +
* <math>\phi_{m}=\prod_{n^2|m}\Phi_{m/n^2}</math>
 +
* as a poly. in <math>x</math>, <math>\deg \phi_m(x,y)=\sigma_1(m)=\sum_{d|m}d</math>
 +
 
 +
;examples
 +
* <math>m=1</math>, <math>\phi_1(x,y)=\Phi_1(x,y)</math>
 +
* <math>m=2</math>, <math>\phi_2(x,y)=\Phi_2(x,y)</math>
 +
* <math>m=3</math>, <math>\phi_3(x,y)=\Phi_3(x,y)</math>
 +
* <math>m=4</math>, <math>\phi_4(x,y) = \Phi_1(x,y)\Phi_4(x,y) = x^7+\dots</math>
 +
* interested in <math>F_m(x):=\phi_m(x,x)\in \Z[x]</math> :
 +
:<math>
 +
F_1(x)=0
 +
</math>
 +
:<math>
 +
F_2(x) = -(x-1728)(x+3375)^2(x-8000) = -H_{4}(x)H_{7}(x)^2H_{8}(x)
 +
</math>
 +
:<math>
 +
F_3(x) = -x(x-8000)^2  (x+32768)^2(x-54000)  = - H_3(x)H_{8}(x)^2H_{11}(x)^2H_{12}(x)
 +
</math>
 +
 
 +
* <math>F_m(x)\neq 0</math> if <math>m</math> is not a perfect square
 +
 
 +
* Hurwitz calculated its degree :
 +
:<math>\deg F_m(x)= \sum_{d|m}\max(d,m/d)</math>
 +
 
 +
* Kronecker : explicit factor. in class poly:
 +
:<math>
 +
F_m(x) =\pm \prod_{t\in \Z,t^2 \leq 4m}\mathcal{H}_{4m − t^2}(x)
 +
</math>
 +
where
 +
:<math>
 +
\mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}}
 +
</math>
 +
* Let <math>h_d</math> 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 <math>m</math> is not a perfect sq., define
 +
:<math>
 +
G(m)= \sum_{t\in \Z,t^2 \leq 4m}h_{4m − t^2}
 +
</math>
 +
Then
 +
:<math>
 +
G(m)=\sum_{d|m}\max(d,m/d)
 +
</math>
 +
 
 +
* this is surprising ; class numbers with different disc. have a linear relation!
 +
* geometric interpretation : <math>\deg F_m(x)</math> = number of intersections of two curves <math>\phi_1(x,y)=x-y=0</math> and <math>\phi_m(x,y)=0</math> in <math>\C^2</math>
 +
* Hurwitz computed this for pairs <math>\phi_{m_1}</math> and <math>\phi_{m_2}</math>
 +
 
 +
;thm (class number relation ver. 2)
 +
Assume that <math>m=m_1m_2</math> is not a perfect square. Let <math>A=\C[X,Y]/\langle \phi_{m_1},\phi_{m_2}\rangle</math>. Then
 +
:<math>
 +
|A|=\sum_{n|\gcd(m_1,m_2)}nG(m/n^2)
 +
</math>
 +
 
 +
 
 +
 
 +
===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}
  
 
==매스매티카 파일 및 계산 리소스==
 
==매스매티카 파일 및 계산 리소스==
34번째 줄: 116번째 줄:
 
* Cohen, Paula. 1984. “On the Coefficients of the Transformation Polynomials for the Elliptic Modular Function.” Mathematical Proceedings of the Cambridge Philosophical Society 95 (3): 389–402. doi:http://dx.doi.org/10.1017/S0305004100061697.
 
* Cohen, Paula. 1984. “On the Coefficients of the Transformation Polynomials for the Elliptic Modular Function.” Mathematical Proceedings of the Cambridge Philosophical Society 95 (3): 389–402. doi:http://dx.doi.org/10.1017/S0305004100061697.
 
* Yui, Noriko. 1978. “Explicit Form of the Modular Equation.” Journal Für Die Reine Und Angewandte Mathematik 299/300: 185–200. http://dx.doi.org/10.1515/crll.1978.299-300.185
 
* Yui, Noriko. 1978. “Explicit Form of the Modular Equation.” Journal Für Die Reine Und Angewandte Mathematik 299/300: 185–200. http://dx.doi.org/10.1515/crll.1978.299-300.185
−
* Herrmann, Oskar. 1975. “Über Die Berechnung Der Fourierkoeffizienten Der Funktion $j(\tau )$.” Journal Für Die Reine Und Angewandte Mathematik 274/275: 187–195. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002190532&IDDOC=253998
+
* Herrmann, Oskar. 1975. “Über Die Berechnung Der Fourierkoeffizienten Der Funktion <math>j(\tau )</math>.” Journal Für Die Reine Und Angewandte Mathematik 274/275: 187–195. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002190532&IDDOC=253998

2020년 11월 12일 (목) 01:59 기준 최신판

개요


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

\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}

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