Modular Symbols

Modular symbols are crucial to many algorithms for computing with modular abelian varieties, because they can be used to construct a finite presentation for the $H_1(X_0(N),\mathbf{Z})$ in terms of paths between elements of $\P ^1(\mathbf{Q}) = \mathbf{Q}\cup \{\infty\}$. They were introduced by Birch [Bir71] and studied by Manin, Mazur, Merel, Cremona, and others.

Let  $\mathfrak{M}_2$ be the free abelian group with basis the set of all symbols $\{\alpha,\beta\}$, with $\alpha, \beta\in\P ^1(\mathbf{Q})$, modulo the three-term relations

$$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0,$$

and modulo any torsion. The group $\GL_2(\mathbf{Q})$ acts on the left on $\mathfrak{M}_2$ by

$$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$

where $g$ acts on $\alpha$ and $\beta$ by a linear fractional transformation. The space $\mathfrak{M}_2(\Gamma_0(N))$ of modular symbols for $\Gamma_0(N)$ is the quotient of $\mathfrak{M}_2$ by the subgroup generated by all elements of the form $x - g(x)$, for  $x \in \mathfrak{M}_2$ and $g$ in $\Gamma_0(N)$, modulo any torsion. A modular symbol for $\Gamma_0(N)$ is an element of this space, and we frequently denote the equivalence class that defines a modular symbol by giving a representative element.

Let $\mathfrak{B}_2(\Gamma_0(N))$ be the free abelian group with basis the finite set $\Gamma_0(N)\backslash \P ^1(\mathbf{Q})$. The boundary map $\delta: \mathfrak{M}_2(\Gamma_0(N))\rightarrow \mathfrak{B}_2(\Gamma_0(N))$ sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$ denotes the basis element of $\mathfrak{B}_2(\Gamma_0(N))$ corresponding to $\beta\in\P ^1(\mathbf{Q})$. The cuspidal modular symbols are the kernel $\mathfrak{S}_2(\Gamma_0(N))$ of $\delta$, and the integral homology $H_1(X_0(N),\mathbf{Z})$ is canonically isomorphism to $\mathfrak{S}_2(\Gamma_0(N))$.

Cremona's book [Cre97, §2.2] contains a concrete description of how to compute $\mathfrak{M}_2(\Gamma_0(N))\otimes \mathbf{Q}$ using Manin symbols, which are a finite set of generators for $\mathfrak{M}_2(\Gamma_0(N))$. In general, the easiest way we have found to compute $\mathfrak{M}_2(\Gamma_0(N))$ is to compute $\mathfrak{M}_2(\Gamma_0(N))\otimes \mathbf{Q}$, then compute the $\mathbf{Z}$-submodule of $\mathfrak{M}_2(\Gamma_0(N))\otimes \mathbf{Q}$ generated by the Manin symbols.