Congruences for the coefficients of the modular function j

Abstract

The functionj(τ) = 12³ J(τ) has a Fourier expansion of the form

$$ {x} + \sum\limits_{n = \left. 0 \right|}^\infty {c(n)x^n ,\,(x = e^{2\pi i\tau } )} $$

where the coefficients c(n) are integers. At the end of Chapter 1 we mentioned a number of congruences involving these integers. This chapter shows how so me of these congruences are obtained. Specifically we will prove that

$$ \begin{array}{*{20}c} {c(2n) \equiv 0(\bmod 2^{11} ),} \\ {c(3n) \equiv 0(\bmod 3^5 ),} \\ {c(5n) \equiv 0(\bmod 5^2 ),} \\ {c(7n) \equiv 0(\bmod 7).} \\ \end{array} $$

The method used to obtain these congruences can be illustrated for the modulus 5². We consider the function

$$ f_5 (\tau ) = \sum\limits_{n = 1}^\infty {c(5n)x^n } $$

obtained by extracting every fifth coefficient in the Fourier expansion of j. Then we show that there is an identity of the form

$$ f_5 (\tau ) = 25\left\{ {a_1 \Phi (\tau ) + a_2 \Phi ^2 (\tau ) + \cdots + a_k \Phi ^k (\tau )} \right\}, $$

(1)

where the a_i are integers and Ф(τ) has a power series expansion in x = e^2πiτ with integer coefficients. By equating coefficients in (1) we see that each coefficient of f₅(τ) is divisible by 25.