Computing the cohomology of Shimura curves in quasi-linear time

Abstract: Computing spaces of modular and automorphic forms is an important problem in algorithmic number theory, with in particular Diophantine applications to generalised Fermat and other equations. The case of Shimura curves was studied by Greenberg and Voight by cohomological methods, allowing them to reduce the problem to linear algebra. Relying on work of Imbert, we leverage topological techniques to obtain linear systems so structured that they can be solved in time quasi-linear in the genus.