Double zeta values, double Eisenstein series, and modular forms of level $2$

Abstract

We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.

Appendix: The double Eisenstein series and the period polynomials in the case of \(\mathrm{SL}_2 (\mathbf{Z})\)

In this appendix we briefly recall the relation described in [9] between the double Eisenstein series and modular forms for $\mathrm{SL}_2 (\mathbf{Z})$.

The double Eisenstein series for $\mathrm{SL}_2 (\mathbf{Z})$ was first defined and studied in [7]:

Its Fourier series is given there as

where $\widetilde{\zeta }(r,s) =(2\pi i)^{-r-s} \zeta (r,s), \widetilde{\zeta }(p)=(2\pi i)^{-p}\zeta (p)$, and

By extending the definition in the case of non-absolute convergence using $q$-series, we showed that the double Eisenstein series satisfy the double shuffle relations (in the form described in [7]), that the space of double Eisenstein series contains the space of modular forms on $\mathrm{SL}_2 (\mathbf{Z})$, and made a connection to the period polynomial by looking at the imaginary parts of the $q$-expansions of $G_{r,s}(\tau )$. Specifically, the imaginary parts are given, like (23), by

$$\begin{aligned} \pi \left( \begin{array}{c} G_{2,k-2}(\tau ) \\ G_{3,k-3}(\tau ) \\ \vdots \\ G_{k-2,2}(\tau ) \end{array} \right) = Q_k^{(1)}\left( \begin{array}{c} \widetilde{\zeta }^ (k-3) g_{3}(q) \\ \widetilde{\zeta }(k-5) g_{5} (q) \\ \vdots \\ \widetilde{\zeta }(3) g_{k-3} (q) \end{array} \right) , \end{aligned}$$

where $Q_k^{(1)}$ is the $(k-3)\times (k/2-2)$ matrix given by

$$\begin{aligned} Q_k^{(1)}=\left( (-1)^i\left( {\begin{array}{c}2j\\ i\end{array}}\right) - (-1)^i \left( {\begin{array}{c}2j\\ k-2-i\end{array}}\right) +\delta _{k-2-i,2j}\right) _{\begin{array}{c} 1\le i\le k-3\\ 1\le j\le k/2-2 \end{array}}. \end{aligned}$$

Rather surprisingly, this contains exactly $Q_k$ as a minor. For example,

Precisely, the $i$-th row of $Q_k$ is the $2i-1$-st row of $Q_k^{(1)}$.

The right kernel of $Q_{k}^{(1)}$ corresponds to the even period polynomials (without constant term) of weight $k$ for $\mathrm{SL}_2 (\mathbf{Z})$, an example being ${}^t(1,-3,3,-1)$ in the right kernel of $Q_{12}^{(1)}$ and the corresponding period polynomial $X^8-3X^6+3X^4-X^2$ of weight $12$. As in Corollary 1, by looking at the constant term of the double Eisenstein series and by using the connection to the period polynomial just mentioned, we obtain the upper bound of the dimension of the space of double zeta values:

$$\begin{aligned} \dim \langle \zeta (r,k-r) \mid 2\le r \le k-1 \rangle _{\mathbf{Q}} \le \frac{k}{2}-1-\dim S_k (1) . \end{aligned}$$

Also, elements in the left kernel of $Q_{k}^{(1)}$ produce expressions of modular forms in terms of double Eisenstein series. By comparing the Fourier coefficients, we obtain certain formulas for Fourier coefficients of modular forms. Let us look at some examples in weight $12$.

As the simplest example, take $(0,0,0,0,1,0,0,0,0)$ in the left kernel of $Q_{12}^{(1)}$. This corresponds to the relation

$$\begin{aligned} 2^7\cdot 3\cdot 5^2\cdot 691\,G_{6,6}(\tau )=2^9\cdot 3^2\cdot 5^2\,{\widetilde{G}}_{12}(\tau )-\Delta (\tau ), \end{aligned}$$

where $\Delta (\tau ) = q\Pi _{n>0} (1-q^n)^{24} = \sum _{n>0} \tau (n) q^n$ is the famous cusp form of weight $12$. Comparing the coefficients of both sides, we obtain

$$\begin{aligned} \tau (n)&= \frac{2}{693} \sigma _{11} (n) + \frac{691}{2^2\cdot 3^2 \cdot 7} \sigma _5 (n) - \frac{691}{2^2\cdot 3^2} \sigma _3 (n) \\&+ \frac{5\cdot 691}{2\cdot 3^2\cdot 11} \sigma _1(n) -\frac{2\cdot 691}{3} \rho _{5,5} (n), \end{aligned}$$

where

Incidentally, the Ramanujan congruence

$$\begin{aligned} \tau (n) \equiv \sigma _{11} (n) \pmod {691} \end{aligned}$$

is clearly seen from this.

Secondly take $(0,0,7,28,0,20,0,0,0)$, which gives the relation

$$\begin{aligned} 2^7\cdot 3^2\cdot 5\cdot 7\cdot 691G_{4,8}(\tau )+2^9\cdot 3^2\cdot 5\cdot 7\cdot 691G_{5,7}(\tau )+ 2^9\cdot 3^2\cdot 5^2\\ \cdot 691G_{7,5}(\tau )=2^5\cdot 3^3\cdot 5\cdot 11\cdot 149{\widetilde{G}}_{12}(\tau ) -\Delta (\tau ) \end{aligned}$$

and the formula

$$\begin{aligned} \tau (n)&=\frac{149}{840}\sigma _{11}(n) -\frac{691}{180}\sigma _7(n) -\frac{11747}{126}\sigma _5(n) +\frac{173441}{360}\sigma _3(n)\\&\quad -\frac{3455}{9} \sigma _1(n) -\frac{2764}{3}\rho _{3,7}(n)-\frac{19348}{3}\rho _{4,6}(n) -\frac{13820}{3}\rho _{6,4}(n). \end{aligned}$$

We may take yet other vectors in the left kernel of $Q_{12}^{(1)}$ (the dimension is $6$) and may deduce similar kind of formulas for $\tau (n)$.

Remark 3

Interestingly enough, the matrix $Q_{k}^{(1)}$ appears when we write “motivic” double zeta values in terms of certain basis elements $f_3,\,f_5,\ldots$ using coproduct structure described in F. Brown’s recent important papers [4, 5]. One of the present authors has found the same relation between triple Eisenstein series and motivic triple zeta values. Or a variant (minor matrix) of $Q_{k}^{(1)}$ appears in the work of Baumard and Schneps [3] on a relation of double zeta values and period polynomials. Each of these should be related with each other, but we have not figured out the exact relations.