Arithmetic of the modular function $j_{1,4}$

Chang Kim; Ja Koo

ArticleOriginal scientific text

Title

Arithmetic of the modular function $j_{1, 4}$

Authors ¹, ¹

Affiliations

Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea

Abstract

We find a generator

j_{1, 4}

of the function field on the modular curve X₁(4) by means of classical theta functions θ₂ and θ₃, and estimate the normalized generator

N (j_{1, 4})

which becomes the Thompson series of type 4C. With these modular functions we investigate some number theoretic properties.

Borcherds, R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., and Wilson, R.A., Atlas of Finite Groups, Clarendon Press, 1985.
Conway, J.H. and Norton, S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339.
Deuring, M., Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
Foster, O., Lectures on Riemann Surfaces, Springer, 1981.
Frenkel, I.B., Lepowsky, J., and Meurman, A., Vertex Operator Algebras and the Monster, Academic Press, Boston, 1988.
Frenkel, I.B., Lepowsky, J., and Meurman, A., A natural representation of the Fischer-Griess monster with the modular function J as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 3256-3260.
Kim, C.H. and Koo, J.K., On the modular function j₄ of level 4, preprint.
Kim, C.H. and Koo, J.K., On the genus of some modular curve of level N, Bull. Austral. Math. Soc. 54 (1996), 291-297.
Kim, C.H. and Koo, J.K., On the modular function $j_{1, 2}$ , in preparation.
Koike, M., On replication formula and Hecke operators, preprint, Nagoya University.
Lang, S., Algebra, Addison-Wesley, 1993.
Lang, S.,, Elliptic Functions, Springer, 1987.
Néron, A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I.H.E.S. 21 (1964), 5-128.
Norton, S.P., More on moonshine, in: Computational Group Theory, Academic Press, London, 1984, 185-195.
Rankin, R., Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977.
Schoeneberg, B., Elliptic Modular Functions, Springer, 1973.
Serre, J.-P. and Tate, J., Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517.
Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Tokyo, 1971.
Thompson, J.G., Some numerology between the Fischer-Griess monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352-353.

Title

Arithmetic of the modular function j1,4

Affiliations

Abstract

Bibliography

Arithmetic of the modular function $j_{1, 4}$