A road to inter-universal Teichmuller theory

According to Mochizuki himself, the essential prerequisites for the IUTeich papers are:

• Semi-graphs of Anabelioids (sections 1 to 6)

• The Geometry of Frobenioids I: The General Theory (complete)

• The Geometry of Frobenioids II: Poly-Frobenioids (sections 1 to 3)

• The Etale Theta Function and its Frobenioid-theoretic Manifestations (complete)

• Topics in Absolute Anabelian Geometry I: Generalities (sections 1 and 4)

• Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms (section 3)

• Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (sections 1 to 5)

• Arithmetic Elliptic Curves in General Position (complete)

While other sources also recommend:

• The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories

• The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve

• A Survey of the Hodge-Arakelov Theory of Elliptic Curves I

• A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

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

• Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions

There's also an introductory paper by Yuichiro Hoshi, but at least for the moment it is avaible in japanese only

• Yuichiro Hoshi, Introduction to inter-universal Teichmüller theory

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).

I'll try to cover the "considerable gap" mentioned in Myshkin's answer, from algebraic geometry on the level of Hartshorne to Mochizuki's work. As a disclaimer, I'll mention that my understanding of Mochizuki's approach is very, very far from satisfactory; however, I believe that it's still possible to figure out some of the more basic prerequisites to Mochizuki's work (which at the moment I'm also still trying to learn about) without entirely understanding Mochizuki's work itself.

"0". Algebraic Number Theory - Aside from algebraic geometry, some basic ideas of modern algebraic number theory are certainly important, such as the local-global principle, and why we might want to develop analogues for p-adic fields of things we are already familiar with in the real and complex case.

Suggested references: Algebraic Number Theory by Jurgen Neukirch or Algebraic Number Theory by J. W. S. Cassels and A. Frohlich.

I. Elliptic Curves - Elliptic curves play a big part in Mochizuki's approach. The abc conjecture is equivalent, via the Frey curve, to the Szpiro conjecture, which concerns elliptic curves. Mochizuki's approach is often said to be (including by Mochizuki himself) analogous to proofs of the Szpiro conjecture for the function field case; first, in his earlier "Hodge-Arakelov Theory", he tries to follow the proof of Szpiro himself (see Minhyong Kim's answer to this question), and in his "Inter-Universal Teichmuller Theory", the proof of Bogomolov and Zhang (see Zhang's paper, and Mochizuki's discussion of the analogy).

In addition, the Tate parametrization of elliptic curves plays an important role in Mochizuki's Inter-Universal Teichmuller Theory. See, for example, Kiran Kedlaya's lecture at the 2015 conference on Inter-Universal Teichmuller Theory at Oxford.

Suggested References: The Arithmetic of Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves by Joseph Silverman

II. Anabelian Geometry - Anabelian geometry is the idea that we can "reconstruct" certain algebraic varieties (called anabelian varieties) from their etale fundamental groups. Before developing Inter-Universal Teichmuller Theory, Mochizuki became well-known for proving that hyperbolic curves (which include, for example, elliptic curves with one point removed, and the projective line with three points removed) are anabelian varieties. Ideas of "reconstruction" figure heavily into Mochizuki's approach; although in what way exactly, I have not yet been able to figure out.

Suggested References: The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves by Hiroaki Nakamura, Akio Tamagawa, and Shinichi Mochizuki

Additionally, I second the recommendation of Fesenko's notes mentioned in Myshkin's answer, as well as notes from this seminar on Kummer Classes and Anabelian Geometry at Vermont.