Double Shuffle Relations of Double Zeta Values and Double Eisenstein Series of Level N

摘要

In their seminal paper "Double zeta values and modular forms" Gangl, Kanekoand Zagier defined a double Eisenstein series and used it to study therelations between double zeta values. One of their key ideas is to study theformal double space and apply the double shuffle relations. They also provedthe double shuffle relations for the double Eisenstein series. More recently,Kaneko and Tasaka extended the double Eisenstein series to level 2, proved itsdouble shuffle relations and studied the double zeta values of level 2.Motivated by the above works, we define in this paper the corresponding objectsat higher levels and prove that the double Eisenstein series of level Nsatisfies the double shuffle relations for every positive integer N. In orderto obtain our main theorem we prove a key result on the multiple divisorfunctions of level N and then use it to solve a complicated under-determinedsystem of linear equations by some standard techniques from linear algebra.