Eadon's unexpected encounter with Peter Scholze, a great mathematician

Below is the exchange. I have yet to finish this page, though, as I need to include JavaScript to display the mathematical symbols correctly.

The original link is here, from Peter Voit's "Not Even Wrong" physics blog. I copied some of that material to this page in case, as often happens, pages, and/or comments, vanish from the web. Mathematics Geometric Langlands comments

Jim Eadon: "Can anyone provide a (hand-wavy) summary for the educated layman (Physics post-grad) of what a Fargues-Fontaine curve is, and what’s special about, e.g. in the context of the Langlands programme?"

Peter Scholze: Jim Eadon, let me try to answer. This paper is about the (local) Langlands correspondence over the $p$-adic numbers $\mathbb Q_p$. Recall that $p$-adic numbers can be thought of as power series $a_{-n}p^{-n} + \ldots + a_0 + a_1 p + a_2p^2 + \ldots$ in the “variable” $p$ — they arise by completing the rational numbers $\mathbb Q$ with respect to a distance where $p$ is small. They are often thought of as analogous to the ring of meromorphic functions on a punctured disc $\mathbb D^*$ over the complex numbers, which admit Laurent series expansions $a_{-n} t^{-n} + \ldots + a_0 + a_1 t + a_2t^2 + \ldots$. More precisely, there is this “Rosetta stone” going back to Weil between meromorphic functions over $\mathbb C$, their version $\mathbb F_p((t))$ over a finite field $\mathbb F_p$, and $\mathbb Q_p$.

However, there is an important difference: $t$ is an actual variable, while $p$ is just a completely fixed number — how should $p=2$ ever vary? In geometric Langlands over $\mathbb C$, it is critical to take several points in the punctured disc $\mathbb D^\ast$ and let them move, and collide, etc. What should the analogue be over $\mathbb Q_p$, where there seems to be no variable that can vary?

In one word, what the Fargues–Fontaine curve is about is to build an actual curve in which $p$ is the variable, so “turn $\mathbb Q_p$ into the functions on an actual curve”. It then even becomes possible to take two independent points on the curve, and let them move, and collide. With this, it becomes possible to adapt all (well, at least a whole lot of) the techniques of geometric Langlands to this setup.

This idea of “turning $p$ into a variable and allowing several independent points” is something that number theorists have long been aiming for, and is basically the idea behind the hypothetical “field with one element”. I would however argue that our paper is the first paper to really make profitable use of this idea.