Timeline for Szpiro's conjecture for function fields and Mochizuki's approach to the number field case
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2017 at 12:58 | history | edited | Myshkin |
+ elliptic curves tag
|
|
| Aug 28, 2017 at 17:07 | vote | accept | Anton Hilado | ||
| Aug 25, 2017 at 13:04 | comment | added | Pasten | @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. | |
| Aug 25, 2017 at 10:27 | comment | added | Anton Hilado | @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? | |
| Aug 25, 2017 at 1:27 | answer | added | Myshkin | timeline score: 5 | |
| Aug 22, 2017 at 15:51 | comment | added | Pasten | 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). | |
| Aug 22, 2017 at 14:33 | comment | added | nfdc23 | 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. | |
| Aug 22, 2017 at 11:48 | history | asked | Anton Hilado | CC BY-SA 3.0 |