(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual question from all this. see end of the question for a few focused points.)
I have a somewhat open-ended question so I hope it fits the forum. For reference, I got decent background in algebraic geometry (on the level of, say, Hartshorne's text) and I generally tend to be a geometric thinker. More recently I've also been trying to get into number theory, where the geometric intuition is proving extremely useful.
However there is one seemingly basic aspect of the theory that I cannot fit into my mental model, yet becomes crucial in more advanced applications: that is Minkowski theory aka "geometry of numbers". Its premise is to consider the Minkowski space $K_\mathbb{R} := K\otimes_\mathbb{Q}\mathbb{R}$, and embed various sets (mainly fractional ideals) into it as lattices (discrete cocompact subgroups). There also appears to be a multiplicative variant, with the logarithmic embedding. This is used for one in proofs of Dirichlet's two basic theorems of algebraic number theory (finiteness of class groups and the units theorem), but the resulting constructions and quantities (such as discriminants and regulators) are much more far-reaching.
My algebro-geometric instinct is trained to think of $K_\mathbb{R}$ as a sort of fiber over the unique Archimedean place of $\mathbb{Q}$. On the geometry side, this is isomorphic to a coproduct of "formal neighborhoods of" (= completions at) the infinite places of $K$. In analogy to finite places, elements of $K_\mathbb{R}$ are to be thought of as formal expansions of functions around those points at infinity. So far this makes sense. My problem is that I cannot quite make sense of the significance of lattices in this space, nor their covolumes.
For instance when looking at the discriminant and its properties, what I'd like to think about is as a sort of "locus of ramification" of the downstairs space in a branched covering. However in practice, the discriminant is actually defined as a certian "Gram determinant", or equivalently as a covolume of some lattice. In particular it is the lattice $\mathcal{O}_K \subset K_\mathbb{R}$. To me this seems like a discrepancy between the definition and the real usage. A priori, I wouldn't even guess that the covolume should be an integer, let alone a geometrically significant one! Likewise for covolumes of arbitrary fractional ideals.
It is noteworthy that the relation between discriminants and ramifications actually reduces to a local argument. So for any particular choice of basis, we shouldn't think of the discriminant as a mere determinant/volume, but rather a "function" that returns some volume at every finite place. This makes the covolume definition even more confusing, as Minkowski theory goes via the infinite places -- precisely the not finite ones.
Another noteworthy aspect of the construction is that of the different ideal, which is the "upstairs ramification locus". This also has a lattice-theoretic definition that I can't quite make sense of, however this time a more conceptual characterization of the different ideal shows up as the annihilator of the module of relative differentials. The codifferent plays the role of the dualizing sheaf. Once again -- a (relatively) clear geometric meaning obscured behind a lattice-theoreic construction. How do we make sense of this?
Perhaps in some way the discriminant is "counting" (in the sense of measure theory) something that vaguely resembles "global sections" -- this makes stuff like the Minkowski bound oddly reminiscent to various Riemann-Roch kind of statements (hence the appearance of the discriminant really shouldn't be surprising). The fractional ideal, regarded here as a lattice, prescribes some pole-behavior at the finite places, while an appropriate centrally-symmetric subset prescribes pole-behaviors at the infinite places, then we count elements in their intersection. Is there a way to make this analogy more precise?
Many texts on algebraic number theory discuss more generally lattices in arbitrary vector spaces over global fields, see for example chapter III of Serre's Local fields. Serre's text seems to put emphasis on the "invariant factors" of a lattice, which reminds me of the cycles associated to a coherent sheaf, pushing towards a more intersection-theoretic state of mind. Yet this doesn't explain the role of the standard measure / inner product / whatever we use to define covolumes. Neukirch even goes as far as describing "metrized $\mathcal{O}$-modules", which somehow justify the use of Hermitian metrics as the right gadget to take infinite places into account. A lot of geometry-esque ideas can be phrased in terms of those, such as the "arithmetic Riemann-Roch" theorem and the "genus of a number field" (that later shows up again in functional equations!).
Finally, I must at least mention the word "adèles". It seems like, rather than taking the fibers over the Archimedean places as in $K_\mathbb{R}$, the adèlic approach says we can take any places we want, and in fact let's just take all of them. This is a more uniform and satisfying construction. But as far as I can tell, the adèlic proofs at their core are not so different from the usual geometry-of-numbers development. Please correct me if I'm wrong.
I hope all this background helped clarify where I'm at mentally with this material, and what kind of answers I'm looking for. To summarize, here are some more focused questions:
- What is the conceptual reason that the covolume of the ring of integers can detect where an extension ramifies?
- What is a good geometric way to think about elements of the Minkowski space, subsets of the space, and measures of those subsets?
- Are there concrete interpretations of lattices over number fields, perhaps in terms of intersection theory / vector bundles? If so, what would their covolume mean in this context?
- Why are the infinite places in MInkowski theory able to dictate global behavior so well? Even before moving to the full adèles, would it be possible to just pick any finite set of primes and develop a "geometry-of-numbers" relative to them? Would it be able to provide bounds as strong as Minkowski's bound? Would we be able to see the finiteness of the class group through them?
- Are there analogues of Minkowski theory over function fields, or more general algebraic varieties? Do they also use similar mechanisms of lattices & volumes?
