You want to find, starting from your sheaf, a point in the upper halfplane mod SL(2,Z). Equivalently you want to compute its "j-invariant": τ and τ' in the upper halfplane are in the same SL(2,Z) orbit iff j(τ) = j(τ').

So how do we compute the j-invariant?

en.wikipedia.org/wiki/J-invar...

In plain english, we can use the halfplane to model the geometry of a normal distribution (mu \in R, var > 0). The shortest path between two points with the same var but diff mu is an arc. The "midpoint" of that arc will have a var coordinate greater than either end point.

This week my student inquiry was about complex deg. 1 rational functions mapping the upper halfplane onto itself. They decided which generators of the full group do this, using some nice thinking. Then we looked at the UHP version of Escher prints and worked out the hyperbolic metric. #iteachmath

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