Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10224076 | Journal of Number Theory | 2018 | 36 Pages |
Abstract
Modular graph functions associate to a graph an SL(2,Z)-invariant function on the upper half plane. We obtain the Fourier series of modular graph functions of arbitrary weight w and two-loop order. The motivation for this work is to develop a deeper understanding of the origin of the algebraic identities between modular graph functions which have been discovered recently, and of the relation between the existence of these identities and the occurrence of cusp forms. We show that the constant Fourier mode, as a function of the modulus Ï, consists of a Laurent polynomial in y=ÏImÏ of degree (w,1âw), plus a contribution which decays exponentially as yââ. The Laurent polynomial is a linear combination with rational coefficients of the top term yw, and lower order terms ζ(2k+1)ywâ2kâ1 for 1â¤kâ¤wâ1, as well as terms ζ(2wâ2ââ3)ζ(2â+1)y2âw for 1â¤ââ¤wâ3. The exponential contribution is a linear combination of exponentials of y and incomplete Î-functions whose coefficients are Laurent polynomials in y with rational coefficients.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Eric D'Hoker, William Duke,