Rational period functions and indefinite binary quadratic forms, II

Abstract

In I-4] Eichler introduced a generalized abelian integral, now called an Eichler integral, obtained as a (2k l)-fold integral of an entire modular form of positive weight 2k. Under the action of the modular group, the transformation equation of the Eichler integral contains a polynomial of degree 2k 2 which is called the period polynomial, by analogy with the abelian integral. The coefficients of the period polynomial are, essentially, the periods of the corresponding modular form. This is the basis of the beautiful results of Eichler and Shimura [12] on Eichler cohomology. In [7] Knopp further generalized Eichler integrals by introducing modular integrals with period functions that are now rational functions as opposed to being polynomials. In a sequence of papers I-7, 8] Knopp developed the theory of modular integrals. Parson and Rosen 1-11] provided additional examples; but it was Choie I-3] who finally settled the existence question by showing how to construct rational period functions for any integral weight. The object of this paper is to begin the classification of rational period functions of positive weight. This is accomplished by looking at possible irreducible systems of poles, as defined in Sect. 4. If the irreducible system of poles exhibits algebraic symmetry, that is, if the poles occur in conjugate pairs, then the rational period function of weight 2k, k odd, whose pole set is the irreducible system of poles is of the form