Euler complexes and geometry of modular varieties (Q2427027)

In this article, the author continues his investigations [Math. Res. Lett. 4, No. 5, 617--636 (1997; Zbl 0916.11034); ibid. 5, No. 4, 497--516 (1998; Zbl 0961.11040); Duke Math J. 110, No. 3, 397--487 (2001; Zbl 1113.14020)] of the rather mysterious relationship between the mixed Tate motives whose periods are given by the values of depth \(m\) multiple polylogarithms \[ \text{Li}_{n_1,\dots,n_m}(x_1,\dots,x_m)= \sum_{0<k_1<\dots<k_m} \frac{{x_1^{k_1}\cdots x_m^{k_m}}}{{k_1^{n_1}\cdots k_m^{n_m}}} \] at the \(N\)-th roots of unity on one hand and cohomology groups of certain congruence subgroups \(\Gamma_1(m,N) \subset \text{GL}_m ({\mathbb Z})\) on the other hand. In particular, the attention is drawn to the easiest case \(m=2, n_1=n_2=1\), where he provides a motivic interpretation of this link. As explained in [Math. Res. Lett. 4, No. 5, 617--636 (1997; Zbl 0916.11034)], a certain natural tessellation of the upper half plane \(\mathbb H\) by ideal hyperbolic triangles and all its \(SL_2({\mathbb Z})\)-translates give rise to a natural chain complex for every congruence subgroup \(\Gamma\subset SL_2({\mathbb Z})\), the so-called modular complex of \(\Gamma\), of the open part the modular curve \(Y_\Gamma=\Gamma\backslash{\mathbb H}\). Also in this article, the author constructs a natural morphism from the modular complex of \(\Gamma_1(N)\) to the so-called cyclotomic part \([C_2(N) \to\Lambda^2\widehat C_1(N)]\) of the weight 2 motivic complex of Bloch and Suslin \[ \mathcal{B}_2({\mathbb Q}(\mu_N)) \otimes{\mathbb Q}\overset{\delta_2}\longrightarrow \Lambda^2{\mathbb Q}(\mu_N)^*\otimes{\mathbb Q} \] of the cyclotomic field \({\mathbb Q}(\mu_N)\). Here, \(\widehat C_1(N)=C_1(N) \oplus {\mathbb Q}\) and \(C_2(N)\) (resp., \(C_1(N)\)) is the group of cyclotomic units in \({\mathbb Q}(\mu_N)\) (resp., \(p\)-units if \(N =p^r\) is a prime power) tensored with \({\mathbb Q}\). In the present article, a natural morphism of complexes from the modular complex of \(\Gamma(N)\) to the Bloch-Suslin complex of the modular curve \(Y(N)=\Gamma(N)\backslash{\mathbb H}\) is constructed such that the above mentioned map can be obtained by specializing the Bloch-Suslin complex to the cusp at infinity of the modular curve. The main ingredient of the construction of this morphism is the use of so-called double elliptic units \(\theta_E(a_1,a_2,a_3)\in \mathcal{B}_2(k)\otimes{\mathbb Q}\) for an elliptic curve \(E\) over a field \(k\) and rational torsion points \(a_j\in E(k)[N]\) such that \(a_1+a_2+a_3=0\). These units satisfy the relation \[ \delta_2(\theta_E(a_1,a_2,a_3))=\theta_E(a_1)\wedge\theta_E(a_2)-\theta_E(a_1)\wedge\theta_E(a_3) + \theta_E(a_2) \wedge\theta_E(a_3) \] with \(\theta_E(a)\) as the well-known elliptic units of [\textit{A. B. Goncharov} and \textit{A. M. Levin}, Invent. Math. 132, No. 2, 393--432 (1998; Zbl 1011.11043)] for the elliptic curve \(E\) and an \(N\)-torsion point \(a\); eventually, this relation guarantees that one has constructed a map of complexes as desired. The second main achievement here is the study of double elliptic units for CM elliptic curves: Let \(K\) be an imaginary quadratic field of class number \(1\) and \(E\) an elliptic curve over \(K\) with CM by \(O_K\). For each prime ideal \(P \subset O_K\) (prime to the conductor of \(E\)) the author considers the complex (in degrees 1,2,3) \[ C_\bullet(P)=[C_2(P)\overset{\delta_2}\longrightarrow \Lambda^2\widehat C_1(P)\overset{\delta'}\longrightarrow C_1(P)^{\oplus 2}] \] with \(C_1(P)\) (resp., \(C_2(P)\)) the \({\mathbb Q}\)-vector space of elliptic units (resp., double elliptic units) of \(E\) at \(P\)-torsion points and \(\widehat C_1(P)=C_1(P) \oplus {\mathbb Q}\). For \(K = {\mathbb Q}(i)\) and \(K = {\mathbb Q}(\sqrt {-3})\) there is a distinguished Bianchi tessellation of the hyperbolic 3-space \({\mathbb H}_3\) by the \(\text{GL}_2(K)\)-translates of a single geodesic polyhedron. The author constructs a surjective morphism of complexes from a natural chain complex (with 1-, 2- and 3-simplices) of \(\Gamma_1(P)\backslash {\mathbb H}_3\) induced by the Bianchi tessellation onto the complex \(C_\bullet(P)\).