What is your favorite examples of spectral sequences arising naturally in arithmetic geometry? Please explain it in some detail

I posit the following example, in response to your ambiguous question: The coniveau spectral sequence seems to play an important role in 'arithmetic geometry'. One instance is in class field theory for schemes: From W. Raskind's nice survery article "Abelian class field theory of arithmetic schemes" [AMS, 1992, pgs. 100101]: Let $X$ be an arithmetic scheme, $n>0$ invertible on $X$. Then there is a coniveau spectral sequence (in the etale site): $$E^{p,q}_1 = \bigoplus _{ x\in X^{p} } H^{qp} (k(x), \;\mathbb{Z}/n \; (jp)) \Rightarrow H^{p+q} (X, \mathbb{Z}/n \;(j)) $$ Without going into more details, this sequence plays an important role in defining a reciprocity map from a class group of $X$ to abelian fundamental groups. That's all I will say for now in hopes that the above provides for motivation to delve further into studying coniveau, etc. Finally, one of the best articles I have seen on coniveau is by ColliotThélène, Hoobler, and Kahn, "The BlochOgusGabber theorem" which can be found at: http://www.math.jussieu.fr/~kahn/preprints/prep.html It might be nice to have others' remarks/comments on coniveau, but I don't have any precise questions yet. 

