Higher Math: Algebraic Number Theory and Class Field Theory

Pontryagin duals: see Poonen's notes on Tate's thesis, for instance.

When is the (absolute) norm of an ideal actually the norm of an element?

• Principal ideal (may have unit/positivity issues)
• Ideal class is a square (always works!)

How is this related to what (local or global) class field theory has to say about norms of elements?

Connection from Kummer theory to class field theory.

Also cyclotomic extensions...

Understand this comment of KConrad:

For a finite extension of number fields, the story is similar: ramification indices of all prime ideals over the same prime must be the same. Also residue field degrees of all prime ideals over the same prime must be the same. Moreover, there is a converse result in this case: if, for each prime downstairs, all prime ideals above it have equal ramification indices and equal residue field degrees then the extension is Galois, and the hypothesis on ramification is actually not needed to get the conclusion. Over C residue field degrees are all 1, so no information is stored there.

I accidentally found the originating MO question by wondering how unramified maps of Riemann surfaces relate to notion of Galois cover. (Riemann-Hurwitz says...)

Maybe related:

• MO: densities-of-different-splittings-in-non-galois-extension
• MWood: splitting in non-Galois extensions

Sheaves of differentials

Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on).

Want to merge classical and modern perspectives on calculus, tangent/cotangent spaces, etc. See Liu and Vakil for examples, intuition, and pictures. At the beginning, Vakil's "Key fact 21.2.3." on generators and relations of module of differentials (see November 18, 2017 version) is quite nice, with a tight connection to the affine conormal exact sequence:

21.2.D. EXERCISE. Verify Key fact 21.2.3. (If you wish, use the affine conormal exact sequence, Theorem 21.2.12, to verify it; different people prefer to work through the theory in different orders. Just take care not to make circular arguments.)

Qing Liu Chapter 6, Proposition 1.8, relating differentials on towers of fiber spaces (i.e. "affine cotangent sequence" and "affine conormal sequence"). It's interesting that while differentials are (at least in some special cases of interest) easy to compute directly and explicitly by hand (at least intuitively, though checking the universal property may be boring), the proof goes more geometrically through derivations (Yoneda-esque dual perspective, using universal property of the module of relative differential forms). Reflect more deeply on this?

[EDIT after first reading: See also Vakil, "relative cotangent sequence, affine version" and "conormal exact sequence, affine version" for other discussion.]

Could perhaps be proved directly with more yoga, but the derivations sequence is much more transparent than the originals. I suppose it's not too honest to call derivations "purely geometric", but at least elements of the derivation sets are more or less tangent vectors, and the restriction and inducing of derivations can be understood geometrically in that way.

(Technical note: see the algebraic fact that let us conclude exactness from dual exactness: MSE: Dual sequence and its exactness; MSE: Verifying statement of exact sequences; MSE: Hom functors and exactness; MSE: Confusion about a proof: exactness of modules.)

Krasner's lemma and finite extensions of $\mathbb{Q}_p$

See MSE: Intuition for Krasner's Lemma for the key ingredient.

A compactness theorem for the idele group

See BCnrd's notes, Math 676. A compactness theorem for the idele group:

The proof will be uniform across all global fields, and the key to the proof is an adelic replacement for the role of Minkowski's lemma in the classical argument for number fields. In particular, this gives a new proof of Dirichlet's unit theorem and finiteness of class groups for rings of $S$-integers $\mathcal{O}_{K,S}$.

Riemann-Roch over arbitrary fields

See MO: Is there a Riemann-Roch for smooth projective curves over an arbitrary field? or a book on function fields, like Stichtenoth, Algebraic Function Fields and Codes.

Connection to Tate's thesis & Poisson summation

See any good reference on Tate's thesis. See e.g. MSE: Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves? for an immediate online answer.

Dirichlet theorem, Frobenius density theorem, Chebotarev density theorem

See MO: Frobenius density theorem discussion.