TITLE: Using Eichler-Selberg trace formula to compute dimension of modular forms?
QUESTION [11 upvotes]: Is it possible to use Eichler-Selberg trace formula to compute the dimension of modular forms of weight $k$ for $SL(2,\mathbb Z)$? This was computed by classical methods such as Riemann-Roch.

REPLY [4 votes]: An adelic reference for $GL(2, \mathbb{Z})$ is also "Traces of Hecke operators" by Knightly and Li. You choose matrix coefficent at the infinite place and the characteristic function $GL_2(\mathbb{Z}_p)$ times the center, and plug it into the Arthur trace formula. They do that almost for weight $k \geq 3$, but they work with matrix coefficient. This gives the dimension formula. This is likely to be more fruitful for generalization to higher rank or deeper level, where not always a classical treatment is available.