There has been a remarkable discussion going on for the past couple weeks in the comment section of this blog posting, which gives a very clear picture of the problems with Mochizuki’s claimed proof of the Szpiro conjecture. These problems were first explained in the 2018 Scholze-Stix document Why abc is still a conjecture.

In order to make this discussion more legible, and provide a form for it that can be consulted and distributed outside my blog software, I’ve put together an edited version of the discussion. I’ll update this document if the discussion continues, but it seemed to me to now be winding down.

Depending on one’s background, one will be able to get less or more out of trying to follow this discussion, but it seems to me that it makes an overwhelmingly convincing case that Mochizuki’s articles do not contain a proof of the conjecture and should not be published by PRIMS. No one involved in the discussion claims that there is an understandable and convincing proof in the articles. The discussion is rather about Scholze’s argument that there is no way that the kind of thing Mochizuki is doing can possibly work. While Scholze may not have a fully rigorous, loophole-free argument (and given the ambiguous nature of many of Mochizuki’s claims, this may not be possible) the burden is not on him to do this.

To justify the PRIMS decision to publish the proof, one needs to assume that the referees have some understood and convincing counterargument to that of Scholze, one that nobody has made publicly anywhere. If this really is the case, the editors of PRIMS need to make public these counterarguments, and those mathematicians who find them convincing need to be able to explain them.

A note on comments: if someone has further technical comments on the mathematical issues being discussed at the earlier posting, they should be submitted there. For discussion of issues surrounding publication of the claimed Mochizuki proof, this would be the right place (and I’ve moved a couple recent ones to here). For comments about Szpiro and his conjecture, the posting about him would be an appropriate one.

Update: I hear that the editors of PRIMS are aware of the recent discussion of the problems with the Mochizuki proof, but have decided to go ahead with the publication of the proof anyway. They do not seem to intend to release any information about their editorial process, in particular what counter-arguments to Scholze’s they considered. In effect, they are taking the stand that they have convincing evidence that Scholze is wrong about the mathematics here, but cannot make it public for confidentiality reasons.

Note that the discussion in the comment thread itself has some later entries after the ones gathered in the pdf document I created.