From quadratic polynomials and continued fractions to modular forms

Paloma Bengoechea*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 the Fourier expansion of the kernel function for the 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 in terms of the infinite set of transformations occurring in a continued fraction algorithm of the real variable. We deduce the exponential convergence of the sums, which was conjectured by Zagier as well as one of the three representations.

Original languageEnglish
Pages (from-to)24-43
Number of pages20
JournalJournal of Number Theory
Early online date12 Aug 2014
Publication statusE-pub ahead of print - 12 Aug 2014


  • Binary quadratic forms
  • Continued fractions
  • Modular forms
  • Period polynomials

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