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

• $n=2$인 경우

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

• $n=3$인 경우

$$ \begin{aligned} \Phi_2(x,y) &=x^4+y^4-x^3 y^3+36864000 \left(x^3+y^3\right)-1069956 \left(x^3 y+x y^3\right)+2587918086 x^2 y^2 \\ &+452984832000000 \left(x^2+y^2\right)+8900222976000 \left(x^2 y+x y^2\right)+2232 \left(x^3 y^2+x^2 y^3\right) \\ &-770845966336000000 x y+1855425871872000000000 (x+y) \end{aligned} $$

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxdlhoeW5aTWZqRFk/edit
• http://wstein.org/home/wstein/www/home/mrubinst/phi_l/

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