Generating elements of the annihilating ideal for modular symbols.
It is known that the Eichler-Shimura relations can be used in order to find the vector spaces of modular symbols [see \textit{Yu. I. Manin}, Mat. Sb., N. Ser. 92(134), 378--401, 503 (1973; Zbl 0293.14007); Mat. Sb., N. Ser. 93(135), 621--626, 631 (1974; Zbl 0304.14012); \textit{G. Shimura}, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo (1971; Zbl 0221.10029)]. A similar idea is developed for the group \(\text{ SL}(s,\mathbb Z)\). Generating elements of the annihilating ideal for modular symbols, associated with \(\text{ SL}(s,\mathbb Z)\), are computed. As an application, a computation is given for certain cohomology and homology groups of subgroups of finite index in \(\text{ SL}(s,\mathbb Z)\) (in dimension \(\nu=s(s-1)/2\)) [see \textit{A. Ash}, Math. Ann. 249, 55--73 (1980; Zbl 0438.20035)]. A modular symbol \([M]\) is defined as the class of an integral matrix \(M=(m^1, \dots, m^ s)\) with nonzero \(s\)-columns in the factor module \[ \mathbb Z[{\mathcal M} _ s(\mathbb Z)]/\mathbb Z_ 0[{\mathcal M}_ s(\mathbb Z)], \] where \(\mathbb Z[{\mathcal M}_ s(\mathbb Z)]\) is the free abelian group generated by formal generators \(M\) as above, and \(\mathbb Z_0[{\mathcal M}_ s(\mathbb Z)]\) its subgroup generated by the elements (1) \((a^ 1 m^ 1, \dots, a^ s m^ s)- (m^ 1, \dots, m^ s)\) for all integers \(a^ i\neq 0\); (2) \((m^{\sigma(1)}, \dots, m^ {\sigma(s)})- \text{ sign}(\sigma)(m^ 1, \dots, m^ s)\) for all permutations \(\sigma \); (3) \((m^ 1, \dots, m^ s)\) for all linearly independent \((m^ 1, \dots, m^ s)\); (4) \(\sum_{i=0}^ s (-1)^{i}(m^ 0,\dots,m^{i-1},m ^{i+1},\dots,m^ s)\) for all nonzero \(s\)-columns \((m^1, \dots, m^ s)\).
scientific article; zbMATH DE number 2138544