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divisibility of the Fourier coefficients of Jq functions and the Atkin conjecture for p = 2
 

Summary: On the 2n
divisibility of the Fourier coefficients of
Jq functions and the Atkin conjecture for p = 2
Shigeki Akiyama
College of General Education, Niigata University
Niigata, 950-21, JAPAN
1. Introduction
Let f be the holomorphic modular form of weight 2k, which is a normalized
common eigenform with respect to Hecke operators. Then it is well known that the
Fourier coefficient (n) of f satisfies the equation
(np) - (n)(p) + p2k-1
(n/p) = 0, (1)
for any prime p and any positive integer n. Here (n/p) is defined to be zero when
n/p is not an integer. In [2] and [3], Atkin made a similar conjecture for a modular
function:
Conjecture (Atkin).
Let j(z) be the modular invariant:
j(z) =
n-1
c(n)xn

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics