Update (12 December, 2015): I've written a brief summary of the Oxford workshop on IUTT rather rapidly, so as to save people the trouble of circulating rumours. This seemed to be a reasonable place to put it. All errors in it are my own: http://people.maths.ox.ac.uk/kimm/papers/iutt=clay.pdf Apology: Sorry. I realised right after posting that this note contains something close to a quote from Mochizuki. So I thought I had better check with him first. I'll put the link back soon. --------------------------- After much hesitation, I've decided to post an answer to this question. Rather, the answer is not to this question, but a flawed response to this as well as two previous entries on MO: http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 http://mathoverflow.net/questions/195353/what-is-a-frobenioid I've put it here because this was the most recent post. On the other hand, I am giving something of a physicist's answer to this question as well by directing attention to how the etale theta is used, rather than what it is. It might be reasonable to put on record a strong disclaimer. I have by no means even begun to understand the papers (with the corollary that I can hardly vouch for its correctness). In fact, my understanding is not much deeper than when I wrote the first superficial answer three years ago. Perhaps the state of knowledge can be best summarised by an honest appraisal of the amount of time invested. Over the last three years, I estimate a total of 4 or 5 days looking through the papers with a modicum of concentration, and perhaps a total of 10 hours conversing with Mohamed Saidi and Shinichi Mochizuki. Essentially all of this time, instead of studying systematically, I was doing what most experienced (and lazy) mathematicians do in such a situation: reading a bit, asking a bit, and then sporadically trying to figure out myself what might be going on. Nonetheless, the <a href="http://www.claymath.org/events/iut-theory-shinichi-mochizuki" >Oxford workshop</a> on Interuniversal Teichmueller Theory is fast approaching. So last week, I felt obligated to begin some mathematical preparations myself. For this, I went to Exeter to talk to Saidi on Thursday and Friday, and then had a Skype conversation with Mochizuki yesterday. At this point, I decided it might not be a bad idea to summarise my hypotheses in a short exposition. This is what I recommend to my Ph.D. students: As soon as possible, try to describe a mathematical theory or result in one's own words, faulty as they may be. This is the spirit in which my 'answer' should be read. In fact, I didn't check either with Saidi or Mochizuki if what I wrote is even halfway reasonable, for fear that careful re-examination would send me into an infinite regress and/or paralysis. There are parts that are already sloppy or that I know to be a bit wrong, but it seemed rather painful to try to get it right just now. Needless to say, any serious misunderstanding is my own fault. It's entirely possible that I'll produce revisions both before and after the workshop. However, they will be posted in addition to what I've written now, which will remain available so other people might benefit from taking note of my errors. With that long prologue out of the way, here it is: http://people.maths.ox.ac.uk/kimm/papers/pre-iutt.pdf ---------------- 17/11/2015: A correction about reconstruction of theta values has been added.