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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