The double logarithm and Manin's complex for modular curves
The double logarithm is \(Li_{1,1}(x,y)= \sum_{0< k_1< k_2} x^{k_1} y^{k_2}/k_1 k_2\), a special case of the multiple polylogarithm \(\sum_{0< k_1<\cdots< k_m} x^{k_1}_1\cdots x^{k_m}_m/k^{n_1}_1\cdots k^{n_m}_m\). The double logarithm is related to the dilogarithm by \[ Li_{1,1}(x,y)= Li_2 \Biggl({xy- y\over 1-y}\Biggr)- Li_2 \Biggl({y\over y- 1}\Biggr)- Li_2(x,y), \] which can be verified by differentiation, and the symmetry relation \(Li_{1,1}(x,y)+ Li_{1,1}(y, x)= \log(1- x)(1-y)- Li_2(xy)\) is equivalent to Abel's identity for the dilogarithm. The paper explores a `higher cyclotomy theory'. In the classical case of the cyclotomic field \(\mathbb{Q}(\zeta_p)\), the quotient of the group of units by the cyclotomic units is a finite group of order \(h^+_p\) and the cyclotomic units are generated by \(L_1(\zeta^\alpha_p)= -\log(1-\zeta^\alpha_p)\). The weight 2 analogue of the group of units is \({\mathcal L}(p)_2\) which is the space generated by all 2-framed mixed Tate modules over the scheme \(\text{Spec }\mathbb{Z}[x]/(x^p- 1)\) and the analogue of the subgroup of cyclotomic units is \({\mathcal C}(p)_2\) which is the subspace generated by the motives \(\widetilde Li_{1,1}(x,y)\) \((x^p= y^p=1)\). The author explains a construction which yields a canonical isomorphism of \({\mathcal L}(p)_2/{\mathcal C}(p)_2\otimes \mathbb{Q}\) to \(H_1(X_1(p),\mathbb{Q})_+\), where \(X_1(p)\) is the modular curve. Further papers dealing with the general case are foreshadowed.
scientific article; zbMATH DE number 1117485