From continued fractions and quadratic functions to modular forms

Abstract

In this paper we study certain real functions defined in a very simple way by Zagier as sums of infinite powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. We prove two conjectures of Zagier showing that the sums converge exponentially. We also prove unexpected results on the representation of these functions as sums over simple or reduced quadratic forms and the positive or negative continued fraction of the variable. These arise from more general results on polynomials of even degree. Especially we give the even part of the Eichler integral on a real number x of any cusp form for PSL(2,Z) in terms of the even part of its period polynomial and the continued fraction of x.