From continued fractions and quadratic functions to modular forms

TL;DR

This paper explores special real functions linked to modular forms, proving convergence of certain sums, and revealing new representations involving quadratic forms and continued fractions, deepening understanding of their structure.

Contribution

It proves two conjectures of Zagier on the exponential convergence of these functions and introduces novel representations using quadratic forms and continued fractions.

Findings

Proved exponential convergence of Zagier's sums.

Established new links between these functions and quadratic forms.

Expressed Eichler integrals in terms of period polynomials and continued fractions.

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.