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The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the most prominent replies has been given here and consists simply of a pointer to what is maybe the one single place that sets out to produce a table listing some key statements, namely Poonen's lecture notes (pdf, see section 2.6).

While such a table is much needed (as witnessed conclusively by 35 upvotes, and counting) it seems to me that there'd eventually be much more to put in it in order to do any justice to the topic. Notably the table ought to have a third column besides arithmetic geometry over number fields and function fields, namely the column for complex curves/Riemann surfaces, which brings the geometric Langlands correspondence into the picture. And it would be good for such a table to be hyperlinked, since there'd be so much to say on each single one of its entries.

In short, that motivated me to start to try to compile a

on the $n$Lab. I got somewhere, but there is still some way to go. I have some questions, too. (And I would like to stress that nothing in this table is meant as claim of mine, all is me trying to reproduce what is known. If there is anything that seems outrageous, then this is a mistake on my part and I will do my best to fix it.)

So in general one question is: does this look about right? And: what seem to be glaring omissions. (I am aware of some, but I hope to hear of those that I am not aware of yet.)

But I also have this slightly more concrete question:

From one point of view the search for $\mathbb{F}_1$ is the search for a systematic theory that would promote the function field analogy from an analogy to a well controled base change away from $\mathrm{Spec}(\mathbb{F}_1)$. That idea is expressed for instance here in another previous MO discussion of this point. However, when I scan the literature on $\mathbb{F}_1$ then I see plenty of discussion of zeta functions over these various bases, but little about the bulk rest of the function field analogy table. Does existing $\mathbb{F}_1$-theory have much to say here? For instance the very first line of the (either!) function field analogy table states that $\mathbb{Z}$ is analogous to $\mathbb{F}_q[x]$ and to some extent so "as $q \to 1$". So from the point of view of the function field analogy it would seem that the central request on any theory of $\mathbb{F}_1$ would be to give rise to a truth that reads in symbols like "$\mathbb{Z} \simeq \mathbb{F}_1[x]$", whatever it is that makes this true. I see that some people do expect just this from a theory of $\mathbb{F}_1$ (for instance in the first footnote here). However, what I have seen as actual proposals for $\mathbb{F}_1[x]$ seems to be headed in a rather different direction. Unless I am missing something, of course, and my question is: am I? Which approach to $\mathbb{F}_1$ should I look at for function field analogy purposes beyond (and that probably means: prior to) zeta functions?

Finally to close an already long and vague question with something even broader, just for those who might enjoy it (all others please stop reading): what I am eventually after is an answer to my old MO question p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf). Namely there are so many hints already in string theory that a function field analogy base-changing us from complex curves to arithmetic geometry over $\mathbb{F}_1$ plays a role, that I'd like to have a good enough mathematical theory of the analogy that would allow to put these hints together to a nice statement. For instance looking at application or mirror symmetry to geometric Langlands as in Gerasimov-Lebedev-Oblezin 09 makes one want to ask: "What is a sigma model in $\mathbb{F}_1$-geometry?" I am wondering how far $\mathbb{F}_1$-theory -- or maybe global analytic geometry? -- may have gotten in this respect, or what the prospect seems to be. Is this even in the line of sight of present research into $\mathbb{F}_1$? If not: what is, if anything?

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    $\begingroup$ This is not in the direction of your question, but I think there are some difficulties with the analogy when considering arithmetic groups. The patterns in classification of finite subgroups in $GL_n\mathbb{Z}$ and $GL_n\mathbb{F}_p[T]$ seem to differ in a way that is not clear to me. Also, $S$-arithmetic groups in the number field case are $FP_\infty$, but they only have bounded finiteness properties in the function field case. $\endgroup$ – Matthias Wendt Jul 22 '14 at 20:11
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    $\begingroup$ Not Weil zeta function? $\endgroup$ – Will Sawin Jul 22 '14 at 20:25
  • $\begingroup$ @Matthias, thanks for the comment. To my mind, part of the output of a good theory of absolute geometry should be to tell which analogies we should expect to see in the first place. To me the fact, for instance, that number fields have a genus which behaves in theorems just as the genus of a curve does is enough to conclude that there myst be global theory. Whether this will make combinatorics of finite subgroups be analogous I find secondary, for that seems much less conceptually compelling than, say, the arithmetic genus of a number field. $\endgroup$ – Urs Schreiber Jul 22 '14 at 22:56
  • $\begingroup$ @WillSawin You mean Artin zeta function but, yes, it's a glaring omission. $\endgroup$ – Felipe Voloch Jul 23 '14 at 4:36
  • $\begingroup$ @WillSawin, right, thanks, fixed now! $\endgroup$ – Urs Schreiber Jul 23 '14 at 10:14
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I think the biggest thing missing from this table is the geometric picture over function fields. Almost everything under "complex Riemann surface" on your table makes sense over for algebraic curves over an arbitrary field.

But of course if you go so far as to split the function field case into two columns, the first and second column of your table would be almost identical, as would the third and fourth column, which would be wasteful.

However this does represent how the function field analogy actually works. Usually passing from number fields to function fields is a simple bookkeeping step (changing notation for the same concepts), and moving geometry on curves from one field to another is again bookkeeping, but passing from arithmetic to geometry over a single field requires some insight, although usually a simple one.

Second, I think many mathematicians working in number theory are skeptical of $\mathbb F_1$-theory and prefer to keep it an analogy. One reason is that the analogy can fail if you look in the wrong places. If $\mathbb Z= \mathbb F_1[t]$, what is $\mathbb F_1[t^2]$? Also consider the zeta functions of these fields. The Riemann zeta function has infinitely many zeroes, while the zeta function of $\mathbb F_q(t)$ has none. To study the zeros of the Riemann zeta function in the function field model, mathematicians pass to the limit of infinitely large $g$. James Borger also pointed out the issues with the discriminant.

So clearly in translating questions between the function field and number field worlds some discretion is necessary.

Similar to this all currently existing $\mathbb F_1$-theories have some kind of problem where they don't fit with our intuitive idea of what an $\mathbb F_1$-theory should look like - in fact I believe that is known to be contradictory. Certainly good mathematical work can be done by finding clever workarounds and deftly avoiding these problems, and it might go so far as to solve completely or make progress on otherwise intractable number field problems.

Third let me say that, not knowing much about string theory, it seems to me that your other question should not run into these problems. Indeed you seem to be concerned only with (various sophisticated forms of) analysis and integration over these fields. Identities of integrals and things like that tend to translate very well among different contexts once you've found the right way of looking at them - I have seen many examples of this. But I don't have any idea what to do in your particular problem.

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