- QUOTIENTS OF THETA SERIES AS RATIONAL FUNCTIONS OF j(sub)1,8
- ㆍ 저자명
- Hong. Kuk-Jin,Koo. Ja-Kyung
- ㆍ 간행물명
- Journal of the Korean Mathematical Society
- ㆍ 권/호정보
- 2001년|38권 3호|pp.595-611 (17 pages)
- ㆍ 발행정보
- 대한수학회
- ㆍ 파일정보
- 정기간행물| PDF텍스트
- ㆍ 주제분야
- 기타
Let Q(n,1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[X] in Q(n,1), the theta series $ heta$(sub)A(z) = ∑(sub)X∈Z(sup)n e(sup)$pi$izA[X] (Z∈h (※Equations, See Full-text) the complex upper half plane) is a modular form of weight n/2 for the congruence group Γ$_1$(8) = {$delta$∈SL$_2$(Z)│$delta$≡()mod 8} (※Equation, See Full-text). If n$geq$24 and A[X], B{X} are tow quadratic forms in Q(n,1), the quotient $ heta$(sub)A(z)/$ heta$(sub)B(z) is a modular function for Γ$_1$(8). Since we identify the field of modular functions for Γ$_1$(8) with the function field K(X$_1$(8)) of the modular curve X$_1$(8) = Γ$_1$(8)\h(sup)* (h(sup)* the extended plane of h) with genus 0, we can express it as a rational function of j(sub) 1,8 over C which is a field generator of K(X$_1$(8)) and defined by j(sub)1,8(z) = $ heta$$_3$(2z)/$ heta$$_3$(4z). Here, $ heta$$_3$ is the classical Jacobi theta series.