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

• Autorzy: Chang Heon Kim , Ja Kyung Koo
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 14H55: Riemann surfaces; Weierstrass points; gap sequences
• 11R04: Algebraic numbers; rings of algebraic integers
• 11R37: Class field theory
• 11F11: Holomorphic modular forms of integral weight
• 11F03: Modular and automorphic functions
• 11F22: Relationship to Lie algebras and finite simple groups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 84
• Numer: 2
• Strony: 129-143

Daty

wydano

1998

otrzymano

1996-12-06

poprawiono

1997-04-01

Twórcy

autor

Chang Heon Kim

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

autor

Ja Kyung Koo

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

Bibliografia

• [1] Borcherds, R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
• [2] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., and Wilson, R.A., Atlas of Finite Groups, Clarendon Press, 1985.
• [3] Conway, J.H. and Norton, S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308-339.
• [4] Deuring, M., Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
• [5] Foster, O., Lectures on Riemann Surfaces, Springer, 1981.
• [6] Frenkel, I.B., Lepowsky, J., and Meurman, A., Vertex Operator Algebras and the Monster, Academic Press, Boston, 1988.
• [7] 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.
• [8] Kim, C.H. and Koo, J.K., On the modular function j₄ of level 4, preprint.
• [9] 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.
• [10] Kim, C.H. and Koo, J.K., On the modular function $j_{1,2}$, in preparation.
• [11] Koike, M., On replication formula and Hecke operators, preprint, Nagoya University.
• [12] Lang, S., Algebra, Addison-Wesley, 1993.
• [13] Lang, S.,, Elliptic Functions, Springer, 1987.
• [14] Néron, A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. I.H.E.S. 21 (1964), 5-128.
• [15] Norton, S.P., More on moonshine, in: Computational Group Theory, Academic Press, London, 1984, 185-195.
• [16] Rankin, R., Modular Forms and Functions, Cambridge Univ. Press, Cambridge, 1977.
• [17] Schoeneberg, B., Elliptic Modular Functions, Springer, 1973.
• [18] Serre, J.-P. and Tate, J., Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517.
• [19] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Tokyo, 1971.
• [20] Thompson, J.G., Some numerology between the Fischer-Griess monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), 352-353.