Not that I understand anything of this, but there is the following paper by Peter Scholze that seems to be in-topic here:

On torsion in the cohomology of locally symmetric varieties

Anybody is welcome to edit this answer adding explanations or insights.

Not that I understand anything of this, but there is the following paper by Peter Scholze that seems to be in-topic here:

• On torsion in the cohomology of locally symmetric varieties, Annals of Mathematics 182 issue 3 (2015) 945-1066 doi:10.4007/annals.2015.182.3.3, arXiv:1306.2070.

Anybody is welcome to edit this answer adding explanations or insights.