Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's answer to this MO question?

I am looking at this in the context of Mochizuki's much-discussed approach to an analogue of the same conjecture for number fields, so I am particularly looking for an emphasis on concepts such as the Gauss-Manin connection and the importance of finding an arithmetic analogue.

As for my relevant background, I know some basic algebraic geometry, including cohomology and elliptic curves, but I have very little knowledge of subjects such as deformation theory and Hodge theory (which I am assuming these topics belong to). I know about connections in the context of differential geometry. If there are any relevant prerequisites it would be very helpful to have them enumerated.