An introduction to computing modular forms using modular symbols

Summary

We explain how weight-two modular forms on 0.N/ are related to modular symbols, and how to use this to explicitly compute spaces of modular forms.

The definition of the spaces of modular forms as functions on the upper half plane satisfying a certain equation is very abstract. The definition of the Hecke operators even more so. Nevertheless, one wishes to carry out explicit investigations involving these objects.

We are fortunate that we now have methods available that allow us to transform the vector space of cusp forms of given weight and level into a concrete object, which can be explicitly computed. We have the work of Atkin–Lehner, Birch, Swinnerton-Dyer, Manin, Mazur, Merel, and many others to thank for this (see, e.g., [Birch and Kuyk 1975; Cremona 1997; Mazur 1973; Merel 1994]). For example, we can use the Eichler–Selberg trace formula, as extended in [Hijikata 1974], to compute characteristic polynomials of Hecke operators. Then the method described in [Wada 1971] gives a basis for certain spaces of modular forms. Alternatively, we can compute ⊝-series using Brandt matrices and quaternion algebras as in [Kohel 2001; Pizer 1980], or we can use a closely related geometric method that involves the module of enhanced supersingular elliptic curves [Mestre 1986]. Another related method of Birch [1991] is very fast, but gives only a piece of the full space of modular forms. The power of the modular symbols approach was demonstrated by Cremona in his book [1997], where he systematically computes a large table of invariants of all elliptic curves of conductor up to 1000 (his online tables [Cremona undated] go well beyond 100,000).