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I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf). I'm trying to understand why the amalgamated sum of ...

I couldn't find a demonstration of this theorem: Given $A \in SO_2(\mathbb{Z}[{1 \over q_1},\dots,{1 \over q_k}])$ and $p$ prime $\notin \{q_1,\dots,q_k\}$ $\exists n \in \mathbb{N} : A^n=Id$ and $A\...

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \...