Introduction#
This chapter, which was written by William Stein (wstein@gmail.com) with the help of feedback from Kevin Buzzard, describes how to compute with modular symbols using Magma. Modular symbols provide a presentation for certain homology groups, and as such they can be used to compute an eigenform basis for spaces of cusp forms \(S_k(N,\varepsilon)\), where \(k\geq 2\) is an integer and \(\varepsilon\) is an arbitrary Dirichlet character. Their generality makes modular symbols a natural tool in applications ranging from verification of modularity of Galois representations to elliptic curve computations.
Our implementation of modular symbols algorithms in Magma was deeply influenced by [Cremona, 1992, Cremona, 1997, Merel, 1994]. The algorithms for computing arithmetic invariants of modular abelian varieties are based on [Stein, 2000]. Those unfamiliar with modular symbols might wish to consult [Stein, 2008] and the references contained therein and peruse [Frey and Muller, 1999].
Modular Symbols#
The modular group \({\operatorname{SL}}_2({\mathbb{Z}})\) is the group of \(2\times 2\) integer matrices with determinant \(1\). For each positive integer \(N\) let \(\Gamma_0(N)\) denote the subgroup of \({\operatorname{SL}}_2({\mathbb{Z}})\) of matrices that are upper triangular modulo \(N\). As explained in the survey paper [Diamond and Im, 1995], there is an algebraic curve \(X_0(N)\) over \({\mathbb{Q}}\) attached to \(\Gamma_0(N)\). The Riemann surface attached to \(X_0(N)\) is a compactified quotient of the upper half plane by the action of \(\Gamma_0(N)\) via linear fractional transformations. Modular symbols provide an explicit computable presentation for certain “(co-)homology groups” attached to modular curves \(X_0(N)\).
Let \(\bf P^1({\mathbb{Q}})\) denote the set \({\mathbb{Q}}\cup \{\infty\}\), and fix a field \(F\). Let \(\bf M\) denote the \(F\)-vector space generated by the formal symbols \(\{a,b\}\), with \(a,b \in \bf P^1({\mathbb{Q}})\), modulo the relations \(\{a,b\} + \{b,c\} + \{c,a\}=0\) for all \(a,b,c\in {\mathbb{Q}}\). (The symbol \(\{a,b\}\) can be visualized as the homology class of a geodesic path from \(a\) to \(b\) in the upper half plane.) Fix a positive integer \(k\). A weight-\(k\) symbol is a formal product \(X^iY^{k-2-i}\{a,b\}\), where \(X^iY^{k-2-i}\in F[X,Y]\). Denote by \(\bf M_k\) the formal \(F\)-vector space with basis the set of all weight-\(k\) modular symbols (thus \(\bf M_k \approx \bf M\otimes {\rm Sym}^{k-2}(F\times F)\)). The group \({\operatorname{GL}}_2({\mathbb{Q}})\) acts on the left on \(\bf M_k\); the matrix \(g=\begin{pmatrix}u&v\\ w&z\end{pmatrix}\) in \({\rm GL}_2({\mathbb{Q}})\) acts by
A mod \(N\) Dirichlet character \(\varepsilon\) is a homomorphism \(\varepsilon:({\mathbb{Z}}/N{\mathbb{Z}})^*\rightarrow F^*.\) The vector space \(\bf M_k(N,\varepsilon;F)\) of modular symbols of weight \(k\), level \(N\) and character \(\varepsilon\) over \(F\) is the quotient of \(\bf M_k\) by the subspace generated by all \(x - \varepsilon(u)g(x)\), for \(x\) in \(\bf M_k\) and \(g=\begin{pmatrix}u&v\\ w&z\end{pmatrix}\in \Gamma_0(N)\). We denote the equivalence class that defines a modular symbol by giving a representative element.
The space of modular symbols is a finite-dimensional vector space, and there is a natural finite presentation for it in terms of Manin symbols.