모듈라 곡선 X0(50)

개요

• 대수곡선 $X_0(50)$
• 종수 2
• 곡선 위에 정의된 함수

$$ u=q^{-1}\prod_{n=1}^{\infty}\frac{(1-q^{25n})(1-q^{2n})}{(1-q^{n})(1-q^{50n})} =\frac{1}{q}+1+q+2 q^2+2 q^3+3 q^4 \\ v=q^{-3}\prod_{n=1}^{\infty}\frac{(1-q^{25n})(1-q^{2n})^2}{(1-q^{n})(1-q^{50n})^2} = \frac{1}{q^3}+\frac{1}{q^2}+1+q^3+q^7 $$

• 다음이 성립한다

$$ v+5u^3/v=u^3-2u^2-2u+1 $$

• involution

$$ u\mapsto -1/u, v\mapsto v/u $$

리뷰, 에세이, 강의노트

• Calegari, Frank. 2006. “Book Review: A First Course in Modular Forms.” Bulletin of The American Mathematical Society - BULL AMER MATH SOC 43 (03): 415–422. doi:10.1090/S0273-0979-06-01098-6. http://www.ams.org/journals/bull/2006-43-03/S0273-0979-06-01098-6/S0273-0979-06-01098-6.pdf
• Birch, B. J. "Some calculations of modular relations." In Modular Functions of One Variable I, pp. 175-186. Springer Berlin Heidelberg, 1973.