Where can I find more details on the proof of Szpiro's conjecture for function fields, as mentioned in Minhyong Kim's [answer](https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658) to this MO [question](https://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture)?


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.



Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
You might want to also consider the geometric ("symplectic") version of the conjecture, since Mochizuki alredy has a paper outlining the relationship between Bogomolov's proof and his own IUT theory.


* Shinichi Mochizuki, "[Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichmuller Theory](http://www.kurims.kyoto-u.ac.jp/~motizuki/Bogomolov%20from%20the%20Point%20of%20View%20of%20Inter-universal%20Teichmuller%20Theory.pdf)" (2016)


The original relevant papers are:


* J. Amorós, F. Bogomolov, L. Katzarkov, T. Pantev, I. Smith, "[Symplectic Lefschetz fibrations with arbitrary fundamental groups](https://arxiv.org/abs/math/9810042)" (2000)
* Shouwu Zhang, "[Geometry of algebraic points](https://web.math.princeton.edu/~shouwu/publications/iccm98.pdf)" (2001)


