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From quadratic polynomials and continued fractions to modular forms

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Identifiers
DOI: 10.1016/j.jnt.2014.07.001
Keywords
  • Modular Forms
  • Period Polynomials
  • Binary Quadratic Forms
  • Continued Fractions
Disciplines
  • Computer Science

Abstract

Abstract We study certain real functions defined in a very simple way by Zagier as sums of 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 give three different representations of these sums in terms of a finite set of polynomials coming from reduction of binary quadratic forms and the infinite set of transformations occurring in a continued fraction algorithm of the real variable. From these we deduce the exponential convergence of the sums, which was conjectured by Zagier as well as one of the three representations.

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