The Behavior of General Theta Functions Under the Modular Group and the Characters of Binary Modular Congruence Groups. I

Abstract

The general formulae for the transformation of the Jacobian theta functions under arbitrary modular substitutions were given in 1858 by Hermite.1 Since then several authors have applied Hermito's method to the study of the behaviour of other and more general functions under arbitrary modular substitutions of the variable!2 The same method will be applied in this paper to multiple theta series, formed with an arbitrary positive definite quadratic form in any number of variables, and in which the variables of summation are subject to certain congruence conditions. By a suitable choice of these congruence conditions a set of theta series is obtained with the property that if a modular transformation is applied to one of the series of the set, the result can be represented as a linear combination (with constant coefficients) of the series of the set. The formulae obtained in this way define a matrix representation of the infinite modular group. A matrix representation of modular congruence groups is then obtained by dividing the series of the set by a suitably chosen function. The remaining part of the paper is devoted to the problem of the reduction of this matric representation as a sum of irreducible representations. The modular congruence groups to modulus pX, where p is a prime number, merit special attention. For X = 1 (the binary modular group mod p) the irreducible characters have been determined by Frobenius.3 The irreducible characters of the binary modular group mod p2 have been determined almost at the same time by H. Rohrbach4 and by H. W. Praetorius.5 For X _ 3 the irreducible characters are not yet known. However, the reduction of the above mentioned matric representation by the method of this paper does not presuppose the knowledge of the irreducible characters. On the contrary this reduction can be used to determine the irreducible characters contained in this representation. In the second part of this paper the special case of the binary