KANEKO Masanobu KYUSHU UNIVERSITY Graduate School of Mathematics, Assosiate Professor, 大学院・数理学研究科, 助教授 (70202017)
YOSHIDA Eiji KYUSHU UNIVERSITY Graduate School of Mathematics, Assistant Professor, 大学院・数理学研究科, 講師 (20220626)
SUEYOSHI Yutaka KYUSHU UNIVERSITY Graduate School of Mathematics, Asistant Professor, 大学院・数理学研究科, 講師 (80128040)
|Budget Amount *help
¥5,300,000 (Direct Cost : ¥5,300,000)
Fiscal Year 1997 : ¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1996 : ¥2,700,000 (Direct Cost : ¥2,700,000)
We computed hypergeometric polynomials F (a, b, c ; x) over the finite field Fp for many sets (a, b, c) and primes p. As a result of these computations, we can find 9 sets (a, b, c) of rational numbers which hypergeometric polynomials F (a, b, c ; x) over Fp has striking properties in common. In fact, they corresponds to 9 non-compact arithmetic triangle groups which are commensurable to each other. Their properties in common are as follows : F (a, b, c ; x) can be factorized into linear factors and quadratic factors only. We get a conjecture that the number of linear factors are described as a finite sum of class numbers of imaginary quadratic fields which are determined by the prime p. There are quadratic and higher transformations between these F (a, b, c ; x) over Fp corresponding to the inclusion relation of triangle groups. To prove these results, we used the fact that the roots of some F (a, b, c ; x) = 0 are related to supersingular elliptic curves. For each triangle group, we can find that there exists a special modular function like the j-function with respect to SL_2 (Z) and they play the crutial role in this theory. Moreover we find that the theory of modular forms with respect to these triangle groups can be almost uniformly described through hypergeometric functions. We studied Atkin orthogonal polynomials whose reduction modulo p give supersingular j-polynomials. We can describe these polynomials explicity by hypergeometric functions.