Explicit Computation of Cusp forms via Hecke Action on Cohomology and its Complexity

TL;DR

This paper explores an alternative cohomological method for computing bases of cusp forms on modular curves, analyzing its complexity through an explicit implementation of Wang's algorithm.

Contribution

It provides a detailed complexity analysis of Wang's cohomological approach for computing cusp form bases, offering insights into its computational efficiency.

Findings

The cohomological approach is feasible for explicit computation.

Complexity bounds are established for Wang's algorithm.

The method offers an alternative to modular symbols for cusp form computation.

Abstract

In the literature, the standard approach to finding bases of spaces of modular forms is via modular symbols and the homology of modular curves. By using the Eichler-Shimura isomorphism, a work by Wang shows how one can use a cohomological viewpoint to determine bases of spaces of cusp forms on $\Gamma_0(N)$ of weight $k \geq 2$ and character $\chi$. It is interesting to look at the complexity of this alternative approach, and we do this for an explicit implementation of the algorithm suggested by Wang.