Cohomology of arithmetic groups in GL(2) over definite quaternion algebras

在这个项目中，我们将探索由有理数上的确定四元数代数的序的2 × 2矩阵产生的算术群的上同调。这样的算术群给出了Bianchi群的自然推广，它们充当了双曲5空间的等距。我们将开发计算机程序来计算这些算术群的上同调以及赫克算子的作用。我们将使用这些程序来探索位于现代数论前沿的各种开放猜想。

In this project, we will explore the cohomology of arithmetic groups arising from 2-by-2 matrices with entries coming from orders in definite quaternion algebras over the rational numbers. Such arithmetic groups give natural generalisations of the Bianchi groups and they act as isometries of hyperbolic 5-space. We will develop computer programs that will compute the cohomology of such arithmetic groups together with the action of Hecke operators. We will use these programs to explore various open conjectures that lie at the forefront of modern number theory.