Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지

ARITHMETIC OF THE MODULAR FUNCTION $j_4$

  • Kim, Chang-Heon (Korea Advanced Institute of Science and Technology Department of Mathematics) ;
  • Koo, Ja-Kyung (Korea Advanced Institute of Science and Technology Department of Mathematics)
  • Published : 1999.07.01
Download PDF
⟨ Previous Next ⟩

Abstract

Since the modular curve $X(4)=\Gamma(4)/{\mathfrak{}}^*$ has genus 0, we have a field isomorphism K(X(4)){\approx}\mathcal{C}(j_{4})$ where $j_{4}(z)={\theta}_{3}(\frac{z}{2})/{\theta}_{4}(\frac{z}{2})$ is a quotient of Jacobi theta series ([9]). We derive recursion formulas for the Fourier coefficients of $j_4$ and $N(j_{4})$ (=the normalized generator), respectively. And we apply these modular functions to Thompson series and the construction of class fields.

Keywords

References

  1. Math. Ann. v.185 Hecke operators on Г$_0$(m) Atkin, A.O.L.;Lehner, J.
  2. Invent. math. v.109 Monstrous moonshine and monstrous Lie superalgebras Brocherds, R.E.
  3. Atlas of finites groups Conway, J.H.;Curtis, R.T.;Norton, S.P.;Parker, R.A.;Wilson, R.A.
  4. Bull. London Math. Soc. v.11 Monstrous Moonshine Conway, J.H.;Morton, S.P.
  5. Grundlehren 290 Sphere Packings, Lattices and Groups Conway, J.H.;Sloane, N.J.A.
  6. Analytic Number Theory Goldstein, L.J.
  7. Acta Arith. v.84 Arithmetic of the modular fuction $j_{1,4}$ Kim, C.H.;Koo, J.K.
  8. On the Modular function&j_{1,2}& Kim, C.H.
  9. J. Korean Math. Soc. v.35 On the Modular function&j_4& of level 4 Kim, C.H.
  10. Introduction to Elliptic Curves and Modular Forms Koblitz, N.
  11. On replication formula and Hecke operators Koike, M.
  12. Elliptic Functions Lang, S.
  13. Algebra Lang, S.
  14. Computational group theory More on moonshine Norton, S.P.
  15. Modular Forms and Functions Rankin, R.
  16. Elliptic Modular Functions Schoeneberg, B.
  17. Publ. Path. Soc. Japan no.11 Introduction to the Arithmetic Theory of Automorphic Functions Shimura, G.
  18. Ann. Math. v.97 On modular forms of half-integral weight Shimura, G.
  19. Advanced Topics in the Arithmetic of Elliptic Curves Silverman, J.H.
  20. Bull. London Math. Soc. v.11 Some numerology between the Fischer-Griess monster and the elliptic modular function Thompson, J.G.