Modular symbol ($\{\alpha ,\beta \}$)

A modular symbol is the relative homology class of a path between two modular cusps. Oriented paths satisfy $\{\alpha ,\beta \}+\{\beta ,\gamma \}=\{\alpha ,\gamma \}$. Continued fractions subdivide rational-endpoint paths into unimodular edges, giving a finite presentation through Manin symbols. The Hecke operators act by finite sums of transformed paths, so their matrices can be computed by rational linear algebra.

For a finite-index modular subgroup, a Manin symbol records a coset and its associated oriented unimodular edge. Over the rationals the presentation has the relations $[g]+[gS]=0$ and $[g]+[gST]+[g(ST{)}^2]=0$, for inversion $S$ and translation $T$. They express reversing an edge and the boundary of a triangle. Integral computation also removes torsion caused by elliptic stabilizers. The boundary map sends the symbol to its terminal cusp minus its initial cusp; its kernel is absolute homology.