A compelling selling point for the theory of classical modular forms is the frequent connections between their Fourier coefficients and other areas of mathematics. I will give some examples of this, starting with classical theta functions and ending with new results that relate certain non-holomorphic Siegel modular forms of genus 2 to arithmetic hyperbolic 3-manifolds.

The classical Katok-Sarnak formula relates averages of Maaß forms over Heegner points to certain Fourier coefficients of half integral weight modular forms. This formula is of great structural interest and has applications to equidistribution problems. A generalization to hyperbolic 3 space was, for example, given in the work of Matthes and Mizuno, which is the topic of this talk. We plan to give an accessible introduction to the topic and to sketch a new(ish) proof.

We explain how to describe n-exact structures on a given additive category by use of its functor category. This construction is based on ideas of Enomoto and we discuss the following applications of it: First, every idempotent complete additive category has a unique maximal n-exact structure. This is a higher analogue of results going back to Crivei, Sieg-Wegner, and Rump. Second, our description allows us to explicitly construct new examples of n-exact categories on the category of (graded) projective modules over various classes of algebras including Calabi-Yau algebras and commutative rings.