February 27th
Today I learned about the adeles of $\CC(t),$ which provides some nice analogy to the situation with number fields. Recall geometrically, we are interested in prime ideals as the connection in our analogy: $\CC[t]$ has prime ideals $(t-\alpha)$ for $\alpha\in\CC$ while number fields $K$ have prime ideals $\mf p.$ (There is also the prime $(0).$) We also recall this analogy gave rise to the Zariski topology on $\op{Spec}\mathcal O_K$ by comparing with the Zariski topology on $\op{Spec}\CC[t].$
The individual components of the adeles are local components, and this is clear from $\CC(t).$ Quite literally, studying a rational function $f(t)\in\CC(t)$ "locally'' at the prime $(t-\alpha)$ means we write\[f(t)=\sum_{k=-N}^\infty c_k(t-\alpha)^k\in\CC((t-\alpha))\]as our Taylor series. The term "local'' means that we're studying $f$ by focusing on the individual point at $\alpha.$
I had some confusion in intuition for a while because I felt like "local'' information should still communicate to nearby neighborhoods: studying a rational function at $\alpha$ should give information about $\alpha+\varepsilon.$ However, algebraically this doesn't make much sense, and even analytically, some conditions are needed on $f$ and $\varepsilon$ to make $\alpha$ and $\alpha+\varepsilon$ behave nicely. To be explicit,\[f(t)=\frac{t-(\alpha+\varepsilon)}{t-\alpha}\]has very different behavior at $\alpha$ and $\alpha+\varepsilon.$ It is not clear to me how to prevent this from happening, and I'm not sure if it's easily doable at all. Further, in the number field case, this is more explicit: the prime $(101)$ does not communicate with $(103)$ well.
Anyways, to combine all of our local information together, we define the adeles as\[\sideset{}{'}\prod_{\alpha\in\CC}\CC((t-\alpha)).\]The $\prod'$ refers to the restricted product: we require that all but finitely many of our coordinates in $\CC((t-\alpha))$ to actually live in our "ring of integers'' $\CC[[t-\alpha]].$ Note that rational functions in $\CC(t)$ do embed into these adeles because a rational function only has finitely many poles (say, by Lagrange's theorem on fields). We remark that the ideles can also be defined as\[\sideset{}{'}\prod_{\alpha\in\CC}\CC((t-\alpha))\]where now all but finitely many of the coordinates are in $\CC[[t-\alpha]]^\times.$ We still have $\CC(t)$ embedding into the ideles because a rational function has both finitely many poles and finitely many zeroes. I also think the image of $\CC(t)$ into the ideles is discrete as in the number field case, where\[\prod_{\alpha\in\CC}\CC[[t-\alpha]]^\times\]is our fundamental domain of the ideles modulo $\CC(t).$ I'm not totally sure about this.
The restricted product on the adeles are exactly analogous to wanting all but finitely many of the coordinates of $\AA_K$ to live in $\mathcal O_{K,\nu}$ over our places $\nu.$ We even have the analogy that local fields $K_\nu$ behave as Taylor series like $\CC((t-\alpha)).$ I think what's happening in this analogy is that we think of (fractional) ideals $\mf a$ as functions over primes $\mf p$ with\[\mf a(\mf p)=\mf p^{\nu_\mf p(\mf a)}.\]I should say that I'm trying to think of ideals $\mf a$ as functions of primes $\mf p$ because that's what's going in $\CC$: elements of $\CC(t)$ are effectively rational functions of the primes $(t-\alpha).$
Of course, the provided function doesn't quite catch the behavior we want, but it's catching some: expanding $\mf a$ locally at $\mf p$ like a Taylor series now will maintain the idea of zeroes and poles. Additionally, we see that these primes really do not communicate well with each other.