"J-불변량과 모듈라 다항식"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (→예) |
Pythagoras0 (토론 | 기여) |
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6번째 줄: | 6번째 줄: | ||
==예== | ==예== | ||
+ | * $n=1$ | ||
+ | $$ | ||
+ | \Phi_1(x,y)=x-y | ||
+ | $$ | ||
* $n=2$ | * $n=2$ | ||
$$ | $$ | ||
18번째 줄: | 22번째 줄: | ||
\end{aligned} | \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 | ||
+ | $$ | ||
+ | \mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}} | ||
+ | $$ | ||
+ | * can be also written as a product of $H_d(x)$ | ||
+ | * 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} | ||
+ | |||
+ | |||
==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== |
2019년 3월 26일 (화) 18:29 판
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 $$ \mathcal{H}_d(x) = \prod_{Q\in \mathcal{Q}_d/\Gamma}(x-j(\tau_Q))^{1/w_{Q}} $$
- can be also written as a product of $H_d(x)$
- 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}
매스매티카 파일 및 계산 리소스
- https://docs.google.com/file/d/0B8XXo8Tve1cxdlhoeW5aTWZqRFk/edit
- http://www.math.uwaterloo.ca/~mrubinst/modularpolynomials/phi_l.html
- https://math.mit.edu/~drew/ClassicalModPolys.html
관련도서
- http://books.google.de/books?id=tsTfnHLmgmQC&pg=PA70&dq=157464000000000&hl=de&sa=X&ei=wk0NUvHKO6eK4ATh4oH4BQ&ved=0CDsQ6AEwAQ#v=onepage&q=157464000000000&f=false
- http://books.google.de/books?id=9pUg6nY_-hsC&pg=PA99&dq=157464000000000&hl=de&sa=X&ei=wk0NUvHKO6eK4ATh4oH4BQ&ved=0CDQQ6AEwAA#v=onepage&q=157464000000000&f=false
관련논문
- Sutherland, Andrew V. “On the Evaluation of Modular Polynomials.” arXiv:1202.3985 [cs, Math], February 17, 2012. doi:10.2140/obs.2013.1.531.
- 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
- 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