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I think that is a statement about real closed fields rather than characteristic two phenomena (although it is an interesting observation involving the number 2). [Note: Real closed fields can be characterized by several equivalent properties. One of them is that the algebraic closure is a nontrivial finite degree extension. Another is that a real closed field admits an ordered field structure and any odd-degree polynomial has a root.]
It would help the rest of us if you expanded the text of your question. For example, you might like to tell us a little about motives without log structures, or log structures on varieties over a field. Also, instead of just pointing to a talk and asking for notes, you could start with something like, "Kato gave a talk on motives and log motives. Could someone tell me what one is and why one would study them?"
Incidentally if you use the z -> 2z map to replace \Gamma(2) with the isomorphic group \Gamma_0(4), the map is given by the trace of an element of order 4 (conjugacy class 4A) acting on the monster vertex algebra. It can be expressed (up to a constant shift of 24) as \Delta(2z)^2/(\Delta(z)\Delta(4z)). If you want covering maps for free groups with more generators, you will have to account for moduli, i.e., the maps aren't necessarily taken to each other by Mobius transformations.
