This is a continuation of my previous papers [2], [3], [4] concerning to the monstrous moonshine.

The automorphism group ·O of the Leech lattice L plays an important role in the study of moonshine. Especially it is important to study theta functions associated with quadratic sublattices of L consisting of fixed vectors of elements of ·O. In this paper, we discuss the properties that these functions are expected to satisfy in the relation to the monstrous moonshine.

In [6], Mason reported some connections between sporadic simple group M 24 and certain cusp forms which appear in the ‘denominator’ of Thompson series assigned to Fisher-Griess’s group F1 . In this paper, we discuss the ‘numerator’ of these Thompson series.

In this paper, we study higher reciprocity law of irreducible polynomials f(x) over Q of degree 3, especially, its close connection with elliptic curves rational over Q and cusp forms of weight 1. These topics were already studied separately in a special example by Chowla-Cowles [1] and Hiramatsu [2]. Here we bring these objects into unity.

Let η(z) be Dedekind’s η-function. For any set of integer g = (k 1 …, Ks ), k 1 ≥ k 2 ≥ … ≥ks ≥ 1, put . In this paper, we shall prove McKay’s conjecture which gives some combinatorial conditions about ki on which ηg (z) is a primitive cusp form. As to McKay’s conjecture, we refer [5].

In this paper we shall study relations among the spaces of modular forms mod p attached to Sk(Np, ψχ) and Sk,(N, ψ) by using certain identities between dimensions of these spaces.

Let f(z) be a cusp form of type (l,ε) on Γ0(N) which is a common eigenfunction of all Hecke operators. For such f(z), Deligne and Serre [1] proved that there exists a linear representation

such that the Artin L-function for p is equal to the L-function associated to f(z).

Let N be a positive integer and let Γ 0(N) be the subgroup of SL(2, Z) defined by all matrices with c ≡ 0 (mod N). Let S 2(Γ0(N)) be the space of holomorphic cusp forms of weight 2 with respect to Γ 0(N) and let be the “essential part” of S 2(Γ 0(N)), which is defined in [1].

In this paper we shall prove some congruence relations mod pα between the traces of Hecke operators T(pm) which act on the space of cusp forms of different weights satisfying some congruences mod pα – pα-1 . The method of the proof is very simple and is applicable to all the cases where the trace formula for Hecke operators are already known.