From quadratic polynomials and continued fractions to modular forms

Paloma Bengoechea

doi:10.1016/j.jnt.2014.07.001

From quadratic polynomials and continued fractions to modular forms

Paloma Bengoechea^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

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 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 language	English
Pages (from-to)	24-43
Number of pages	20
Journal	Journal of Number Theory
Volume	147
Early online date	12 Aug 2014
DOIs	https://doi.org/10.1016/j.jnt.2014.07.001
Publication status	E-pub ahead of print - 12 Aug 2014

Keywords

Binary quadratic forms
Continued fractions
Modular forms
Period polynomials

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10.1016/j.jnt.2014.07.001

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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 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.",

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AB - 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.

KW - Binary quadratic forms

KW - Continued fractions

KW - Modular forms

KW - Period polynomials

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SP - 24

EP - 43

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

From quadratic polynomials and continued fractions to modular forms

Abstract

Keywords

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Programme Grant-New Frameworks in metric Number Theory

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From quadratic polynomials and continued fractions to modular forms

Abstract

Keywords

Access to Document

Other files and links

Projects

Programme Grant-New Frameworks in metric Number Theory

Cite this