# Szpiro's conjecture for function fields and Mochizuki's approach to the number field case

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.

• The content in the function field case in characteristic 0 is indeed deformation theory (via the Kodaira-Spencer isomorphism over a moduli stack). A nice short proof based on this is on pp. 2-3 of kurims.kyoto-u.ac.jp/~motizuki/… but this clean argument involves a Deligne-Mumford stack (and assumes the elliptic curve has semistable reduction at bad fibers, to get the map from the complete curve to the proper stack). The prerequisites are Deligne-Rapoport and stacks; it's a long but well-traveled road. – nfdc23 Aug 22 '17 at 14:33
• The usual proof in the function field case can be found in Szpiro's paper "Discriminant et conducteur des courbes elliptiques" in Astérisque 183. This is an important reference that anyone seriously interested on abc should read. However, don't expect this proof to be very analogous to Mochizuki's work. As mentioned in my 2012 answer mathoverflow.net/q/106649 Mochizuki's work is a "$\pi_1$ argument" and, in that sense, it is closer to the argument outlined in my post (cf. Bogomolov et al; Zhang). – Pasten Aug 22 '17 at 15:51
• @Pasten Is this approach also related to the approach involving the Kodaira-Spencer morphism discussed in mathoverflow.net/a/106658/85392 and in Mochizuki's earlier papers on Hodge-Arakelov theory? – Anton Hilado Aug 25 '17 at 10:27
• @AntonHidalgo Yes, the "usual proof in the function field case" appearing in Szpiro's paper that I cited is in fact the one using the KS map. This does not need DM stacks, it's only about elliptic surfaces ---as presented in Szpiro's paper, it is very short and clear. And the other proof that I mentioned (discussed in my old answer mathoverflow.net/q/106649 ) is the same mentioned in Myshkin answer below; this is not about the KS map. – Pasten Aug 25 '17 at 13:04