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http://dx.doi.org/10.4134/JKMS.2007.44.2.343

SUPER-REPLICABLE FUNCTIONS N(j1,N) AND PERIODICALLY VANISHING PROPERTY  

Kim, Chang-Heon (Department of Mathematics Seoul Women's University)
Koo, Ja-Kyung (Korea Advanced Institute of Science and Technology Department of Mathematics)
Publication Information
Journal of the Korean Mathematical Society / v.44, no.2, 2007 , pp. 343-371 More about this Journal
Abstract
We find the super-replication formulae which would be a generalization of replication formulae. And we apply the formulae to derive periodically vanishing property in the Fourier coefficients of the Hauptmodul $\aleph(j_{1,N})$ as a super-replicable function.
Keywords
modular function; Hauptmodul; Thompson series; replicable; super-replicable;
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Times Cited By Web Of Science : 2  (Related Records In Web of Science)
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1 R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), no. 2, 405-444   DOI
2 J. P. Serre and J. Tate, Good reduction of abelian varieties, Ann. Math. (2) 88 (1968), 492-517   DOI
3 D. Alexander, C. Cummins, J. Mckay, and C. Simons, Completely replicable functions, In: Groups, Combinatorics and Geometry, Cambridge Univ. Press (1992), 87-98
4 S. J. Kang, Graded Lie superalgebras and the superdimension formula, J. Algebra 204 (1998), no. 2, 597-655   DOI   ScienceOn
5 A. Neron, Modeles minimaux des varietes abeliennes sur les corps locaux et globaux, Publ. Math. I. H. E. S. 21 (1964), 5-128   DOI
6 C. R. Ferenbaugh, Replication formulae for n/h-type Hauptmoduls, J. Algebra 179 (1996), no. 3, 808-837   DOI   ScienceOn
7 I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer- Griess monster with the modular function J as character, Proc. Nat. Acad. Sci. U. S. A. 81 (1984), no. 10, Phys. Sci., 3256-3260   DOI   ScienceOn
8 I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the monster, Pure and Applied Mathematics, 134, Academic Press Inc., Boston, MA, 1988
9 K. J. Hong and J. K. Koo, Generation of class fields by the modular function $j_{1,12}$, Acta Arith. 93 (2000), no. 3, 257-291
10 N. Ishida and N. Ishii, The equation for the modular curve $X_1$(N) derived from the equation for the modular curve X(N), Tokyo J. Math. 22 (1999), no. 1, 167-175   DOI
11 C. H. Kim and J. K. Koo, Generation of Hauptmoduln of ${\Gamma}_1$(7), ${\Gamma}_1$(9) and ${\Gamma}_1$(10), (preprint)
12 C. H. Kim and J. K. Koo, Self-recursion formulas satisfied by Fourier coefficients of some modular functions, J. Pure Appl. Algebra 160 (2001), no. 1, 53-65   DOI   ScienceOn
13 C. H. Kim and J. K. Koo, Self-recursion formulas of certain monstrous functions, J. Pure Appl. Algebra 171 (2002), no. 1, 27-40   DOI   ScienceOn
14 N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics, 97, Springer-Verlag, 1984
15 C. J. Cummins and S. P. Norton, Rational Hauptmoduls are replicable, Canadian J. Math. 47 (1995), no. 6, 1201-1218   DOI
16 M. Koike, On replication formula and Hecke operators, Nagoya University (preprint)
17 S. Lang, Elliptic Functions, Graduate Texts in Mathematics, 112, Springer-Verlag, 1987
18 J. Mckay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278   DOI
19 M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkorper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272   DOI
20 P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, 1983
21 C. R. Ferenbaugh, The genus-zero problem for n/h-type groups, Duke Math. J. 72 (1993), no. 1, 31-63   DOI
22 S. J. Kang and J. H. Kwon, Graded Lie superalgebras, supertrace formula, and orbit Lie superalgrbra, Prod. London Math. Soc. (3) 81 (2000), no. 3, 675-724   DOI
23 S. J. Kang, C. H. Kim, J. K. Koo, and Y. T. Oh, Graded Lie superalgebras and superreplicable functions, J. Algebra 285 (2005), no. 2, 531-573   DOI   ScienceOn
24 T. Miyake, Modular Forms, Springer-Verlag, 1989
25 C. H. Kim and J. K. Koo, On the genus of some modular curve of level N, Bull. Austral. Math. Soc. 54 (1996), no. 2, 291-297   DOI
26 C. H. Kim and J. K. Koo, Generators of function fields of the moldular curves $X_1$(5) and $X_1$(6), (preprint)
27 C. H. Kim and J. K. Koo, Arithmetic of the modular function $j_{1,8}$, Ramanujan J. 4 (2000), no. 3, 317-338   DOI
28 S. P. Norton, More on moonshine, In: Computational Group Theory, Academic Press (1984), 185-193
29 R. Rankin, Modular Forms and Functions, Cambridge University Press, 1977
30 G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971
31 G. Shimura, On modular forms of half integral weight, Ann. Math. (2) 97 (1973), 440-481   DOI
32 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151, Springer-Verlag, 1994