Notes on class field theory

Appendix A Parting thoughts

Class field theory encompasses a vast expanse of mathematics, so it's worth concluding by taking stock of what we've seen and what we haven't. First, a reminder of the main topics we have covered.

• The Kronecker-Weber theorem: the maximal abelian extension of \(\QQ\) is generated by roots of unity (Theorem 1.1.2).

• The Artin reciprocity law for an abelian extension of a number field (Theorem 2.2.6).

• The existence theorem classifying abelian extensions of number fields in terms of generalized ideal class groups (Theorem 2.2.8).

• The Chebotaryov density theorem, describing the distribution over primes of a number field of various splitting behaviors in an extension field (Theorem 2.4.11).

• Some group cohomology “nuts and bolts”, including the periodicity of Tate cohomology for a cyclic group (Theorem 3.4.1) and Tate's theorem (Theorem 4.3.1).

• The local reciprocity law (Theorem 4.1.2), the local existence theorem (Theorem 4.1.5), and the norm limitation theorem (Theorem 4.1.7).

• The Artin-Tate framework of abstract class field theory, including the abstract reciprocity law (Theorem 5.3.9) and the abstract norm limitation theorem (Corollary 5.3.11).

• Adèles, idèles, and the idelic formulations of the reciprocity law (Theorem 6.4.1) and the existence theorem (Theorem 6.4.2).

• Computations of group cohomology in the local case (multiplicative group; Proposition 4.2.1) and the global case (idèle class group; Theorem 7.1.2, Theorem 7.2.10).

We also gave brief summaries of Brauer groups of number fields (Section 7.6) and of adelic Fourier analysis (Section 6.6).

Now, some things that we haven't covered. When this course was first taught, these topics were assigned as final projects to individual students in the course.

• The Lubin-Tate construction of explicit class field theory for local fields (see [4], IV).

• More details about zeta functions and \(L\)-functions, including the class number formula and the distribution of norms in ideal classes.

• Another application of group cohomology: to computing ranks of elliptic curves via Selmer groups (see [51], X).

• Orders in number fields, and the notion of a ring class field.

• An analogue of the Kronecker-Weber theorem over the function field \(\FF_q(t)\text{,}\) and even over its extensions (see [19]).

• Explicit class field theory for imaginary quadratic fields, via elliptic curves with complex multiplication (see [4], XIII; [10]).

• Quadratic forms over number fields and the Hasse-Minkowski theorem (see [44]).

• Artin (nonabelian) \(L\)-series, the basis of “nonabelian class field theory.”

Some additional topics for further reading would include the following.

• Duality in Galois cohomology, including Tate local duality and Poitou-Tate global duality (see [17]).

• The Golod-Shafarevich inequality and the class field tower problem (see [4], IX).

• Class field theory for function fields (see [47]) and its use to produce curves over finite fields with unusually many points (see manypoints.org¹ and the forthcoming [50]).

• Application of Artin reciprocity to cubic, quartic, and higher reciprocity (see [36]).

• Algorithmic class field theory (see [8], [9]).

• Clausen's \(K\)-theoretic approach to Artin reciprocity (see [7]).

• Higher-dimensional class field theory (see [28]).

• Brauer-Manin obstructions to the existence of rational points on algebraic varieties (see [41], Chapter 8).

And finally, some ruminations about where number theory has gone since the mid-20th century, expanding upon Remark 6.2.13. In its cleanest form, class field theory describes a correspondence between one-dimensional representations of \(\Gal(\overline{K}/K)\text{,}\) for \(K\) a number field, and certain representations of \(\GL_1(\AA_K)\text{,}\) otherwise known as the group of idèles. But what about the nonabelian extensions of \(K\text{,}\) or equivalently the higher-dimensional representations of \(\Gal(\overline{K}/K)\text{?}\)

Building on work of many authors, Langlands has proposed that for every \(n\text{,}\) there should be a correspondence between \(n\)-dimensional representations of \(\Gal(\overline{K}/K)\) and representations of \(\GL_n(\AA_K)\text{.}\) This correspondence is the heart of the so-called Langlands Program, an unbelievably deep web of statements which has driven much of the mathematical establishment for the last few decades. For example, for \(n=2\text{,}\) this correspondence includes on one hand the \(2\)-dimensional Galois representations coming from elliptic curves, and on the other hand representations of \(\GL_2(\AA_K)\) corresponding to modular forms (see [12] for the reinterpretation of the classical theory of modular forms in this language). In particular, it includes the modularity of elliptic curves, proved by Breuil, Conrad, Diamond, and Taylor [2] following on the celebrated work of Wiles [57] and Taylor-Wiles [52] on Fermat's Last Theorem.

Various analogues of the Langlands correspondence have been worked out: for local fields by Harris and Taylor [18], with subsequent simplifications by Henniart [20] and Scholze [43]; and for function fields by L. Lafforgue [29], building on the case \(n=2\) which was treated by Drinfeld. Moreover, one can pin things down better by replacing \(\GL_n\) with a more general algebraic group; in the function fields, this case is addressed by V. Lafforgue [30]. The work of Waldspurger [54] of Laumon and Ngô [34] on the Langlands fundamental lemma is also part of this story.

Recently, some intriguing links have emerged between the Langlands program and some duality theories appearing in mathematical physics, leading to fruitful transfers of ideas in both directions. See [27] for the starting point.

This discussion could continue ad infinitum, so I had to make an arbitrary decision to stop somewhere, and this is that point. Thanks for reading!