According to Mochizuki himself ([here](http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf)), the essential prerequisites for the IUTeich papers are:

* [Semi-graphs of Anabelioids](http://www.kurims.kyoto-u.ac.jp/~motizuki/Semi-graphs%20of%20Anabelioids.pdf) (sections 1 to 6)

* [The Geometry of Frobenioids I: The General Theory](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Frobenioids%20I.pdf) (complete)

* [The Geometry of Frobenioids II: Poly-Frobenioids](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Geometry%20of%20Frobenioids%20II.pdf) (sections 1 to 3)

* [The Etale Theta Function and its Frobenioid-theoretic Manifestations](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Etale%20Theta%20Function%20and%20its%20Frobenioid-theoretic%20Manifestations.pdf) (complete)

* [Topics in Absolute Anabelian Geometry I: Generalities](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20I.pdf) (sections 1 and 4)

* [Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20II.pdf) (section 3)

* [Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf) (sections 1 to 5)

* [Arithmetic Elliptic Curves in General Position](http://www.kurims.kyoto-u.ac.jp/~motizuki/Arithmetic%20Elliptic%20Curves%20in%20General%20Position.pdf) (complete)

While [other sources](http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture) also recommend:

* [The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves.pdf)

* [The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve](http://www.kurims.kyoto-u.ac.jp/~motizuki/The%20Galois-Theoretic%20Kodaira-Spencer%20Morphism%20of%20an%20Elliptic%20Curve.pdf)

* [A Survey of the Hodge-Arakelov Theory of Elliptic Curves I](http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20I.pdf)

* [A Survey of the Hodge-Arakelov Theory of Elliptic Curves II](http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Survey%20of%20the%20Hodge-Arakelov%20Theory%20of%20Elliptic%20Curves%20II.pdf)

Particularly interesting are Fesenko's recent extended remarks on IUT (and learning IUT):

*  Fesenko, [Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions](https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf)

As for the (considerable) gap between Hartshorne and Mochizuki's work, the references on each paper are quite concrete and helpful (see for example the ones on Topics in Absolute Anabelian Geometry I for a good sample).