Remarks on Applications of Class Field Theory

PostCFT.tex 9:09 p.m. October 5, 2013

Remarks on applications of class ﬁeld theory Bill Casselman University of British Columbia [email protected]

Class field theory is ubiquitous in the theory of automorphics forms. I thought it might be a good idea to show several different aspects of how it occurs. The principal theorem about Galois descent is, roughly speaking, that if E/F is a Galois extrension and A an algebraic structure defined over E, then structures over F that become isomorphic to A over E are classified by splittings of a short exact sequence

1 −→ AutE(A) −→ E −→ Gal(E/F) −→ 1 modulo conjugation by elements of AutE(A). Such splittings can be classified by certain cohomology groups 1 H (Gal(E/F ), AutE(A)). Class field theory is invariably required inevaluating such cohomology groups. Contents 1. Galois descent 2. Central simple algebras 3. Weil groups 4. Artin L•functions 5. Langlands’ classification for tori

1. Galois descent

Galois descent is concerned with the question, supposing that I know what some structure is like over a separable closure of a ﬁeld F , what can I say about its structure over F ?

REFERENCES

1. Bill Casselman, ‘Algebraic structures and fields of definition’, available at http://www.math.ubc.ca/ cass/research/pdf/Descent.pdf 2. James Milne, Algebraic geometry, available at http://www.jmilne.org/math/CourseNotes/ag.html Chapter 16 of this book is about descent, mostly Galois but with a brief mention of Grothendieck’s notion of faithfully flat descent. This includes a theory, due to Pierre Cartier, of descent with respect to finite inseparable extensions. 3. Jean•Pierre Serre, Local fields, Springer, 1979. 4. ——, Algebraic groups and class fields, Springer, 1988. This is the standard reference for a translation of Weil’s theory into modern terms.

2. Central simple algebras

Central simple algebras offer the simplest examples of phenomena of descent. The theory also describes the structure of the unit groups of central simple algebras. Central simple algebras are directly related to class ﬁeld theory because equivalence classes of such algebras is in bijection with the fundamental cohomology group 2 × H (Gal(Fs/F ), Fs ).

REFERENCES

5. Max Deuring, Algebren, AMS reprint. Applications of CFT (9:09 p.m. October 5, 2013) 2

This is a very readable book, particularly the last few chapters. But it is in German. 6. Andre´ Weil, Basic number theory (second edition), Springer, 1995. This might be considered the replacement for Deuring’s book, but is much more verbose.

3. Weil groups

I shall look only at local Weil groups.

P-ADIC. Suppose k to be a p•adic field, ks a separable closure. The Galois group Gal(ks/k) projects onto the Galois group of knr/k, which is isomorphic to the Galois group of the residue field extension F/Fq. One can m m define Wk to be the inverse image of powers of the Frobenius F (q = p ). It is dense in Gal(ks/k). There are several advantages to replacing the Galoi group by the Weil group. The principal one is that quotient of Wk by its commutator subgroup, maps into a dense subgroup of the Galois group of the maximal abelian extension of k. The reciprocity theorem for local fields asserts that there is a canonical isomorphism of this × quotient with k . Actually, there are two conventions about this reciprocity map—one has the generator ̟ of p − map to Fm, while the other has it map to the inverse F m. The structure of the Weil group contains within it much information about local class field theory. If E/F is a finite extension, the maximal abelian extension Eab/F is a Galois extension of F . We can define a relative Weil group WE/F as a dense subgroup of Gal(Eab/F ) that fits, by reciprocity, into the short exact sequence

× 1 −→ E −→ WE/F −→ Gal(E/F ) −→ 1 .

One of the basic results of local class field theory is that this extension corresponds to the fundamental class of H2(Gal(E/F ),Ex). In other words, class field theory has something to say about non•abelian extensions of k as well as abelian ones. This result is found originally to [Shafarevitch:1946]. Both [Artin•Tate:1968] and [Tate:1979] explain how the assembly of all Weil groups of p•adic fields encapsulates all of local class field theory for such fields. × × × REAL AND COMPLEX. The Weil group WC is just C . The Weil group WR is the normalizer of C in H , and is generated by Cx and an element σ such that σz = zσ and σ2 = −1. It fits into a short exact sequence

× 1 −→ C −→ WR −→ 1 .

The group WR is therefore the same as WC/R. It does not have any obvious Galois group interpretation, but it does play a Galois•theoretic role concerning certain ﬁelds associated to abelian varieties with complex multiplication.

REFERENCES

7. Emil Artin and John Tate, Class field theory, Benjamin, 1968. 8. Armand Borel and Bill Casselman (editors), Automorphic representations and L•functions, Proceedings of Symposia in Pure Mathematics XXXIII, American Mathematical Society, 1979. 9. I. R. Shafarevitch, ‘On Galois groups of p•adic fields’, Doklady Akademia Nauk 53 (1946), 15–16. There is an English version in his Collected works. Shafarevitch’ paper is the first mention of what later came to be known as the Weil group, but the relationship is not transparent. He constructs it as the normalizer of a torus in a certain p•adic division algebra. This is a very basic result, but I have trouble understanding exactly how it fits in with the general theory. An appendix in the second edition of [Weil:1995] is about this result, but seems to wander around without stating the point clearly. Theorem 6 in [Artin•Tate:1968] is a version of this theorem, but as far as I can see it is not obviously equivalent. 10. John Tate, ‘Number theoretic background’, pp. 3–26 in [Borel•Casselman:1979]. A standard reference for Weil groups, although somewhat dry. Applications of CFT (9:09 p.m. October 5, 2013) 3

4. Artin L•functions

Suppose E/F to be a Galois extension of algebraic number fields, unramified outside the finite set S of primes. Suppose ρ to be a representation of Gal(E/F ). If p is a prime of F , suppose P to be a prime of E over it. The IP iP representation ρ induces a representation ρ of the quotient DP/IP on V . But this quotient is isomorphic to the Galois group of residue fields, generated by what I’ll call FP. The expression

1 I −s det 1 − ρ P (FP)Np is independent of choices, because all are effectively conjugate. I’ll write it as

1 L (s,ρ)= . p I −s det 1 − ρ p (Fp)Np The product 1 L(s,ρ)= Lp(s,ρ)= − . det 1 − ρIp (F )Np s Yp Yp p converges for RE(s) > 1. ∗ If H is a normal subgroup of G and ρ is a representation of G/H, let ρ be the lifting to G. If H is any subgroup of G and ρ a representation of H, let ρ∗ be the induced representation of G.

[three-artins] 4.1. Proposition. We have

(a) Lp(s,ρ + σ)=Lp(s,ρ)Lp(s, σ). ∗ (b) Lp(s,ρ )=Lp(s,ρ). (c) Lp(s,ρ∗)=Lp(s,ρ).

In addition, (d) if ρ is a character then Lp(s,ρ) the same as the L•function defined classically according to the reciprocity theorem. These four properties will be features of other local and global factors of L•functions. Only the last is not straightforward. I follow §3 of [Deligne:1972]. It reduces to a statement about extensions of the finite residue fields. We may assume G to be the cyclic group of order m generated by F, H to be generated by some Fn with n|m. We may first decompose ρ into irreducible one•dimensional components, and then by the sum rule assume ρ itself to be a character. The denominator of the L•function for H is thus now − 1 − ρ(Fn)Np ns = 1 − Xn − with X = ρ(F)Np s.

What about L(s,ρ∗)? We can describe ρ∗(F) explicitly. I recall that ρ∗(g) is the right regular action of G on the space of all functions {f: G −→ C | f(hg)= ρ(h)f)g) for all h ∈ H,g ∈ G} . i We may choose the F with 0 ≤ i

ρ(h) if i = j F (hFj )= i 0 otherwise.

Each coset HFi supports a one•dimension Then the action of F in ρ∗ is

Fi+1 if i

The denomiantor of L(s,ρ∗) is −s det(I − ρ∗(F)Np ) . Applications of CFT (9:09 p.m. October 5, 2013) 4

− It is the characteristic polynomial det(I − MY ) with M = ρ∗(F) evaluated at Y = Np s. It follows from a well known result in linear algebra that this is the same as det(I − ρ(FnY n). If M is any linear transformation of dimension n, its characteristic polynomial is

2 2 det(I − Mt) = 1 − (TR M) t + (TR M) t + ···± det(M) tn . V Suppose n n−1 n−2 P (x)= x + an−1x + an−2x −··· + a0 to be a polynomial of degree n. Multplication by x on C[x] takes

i+1 i x if i

It matrix is 0 0 . . . 0 −a0  1 0 . . . 0 −a1  MP = 0 1 . . . 0 −a2    . . .     0 0 . . . 1 −an−1  The following is a familiar result in linear algebra:

[companion-matrix] 4.2. Lemma. The characteristic polynomial of the matrix MP is P . i The matrix MP is called the companion matrix of P . This Lemma is a statement about the trace of M, and may be proved by induction, moving from lower right to upper left. V For example, you can see easily the characteristic polynomial of

0 0 −a0 M =  1 0 −a1  0 1 −a2   3 2 is t + a2t + a1t + a0.

THE FUNCTIONAL EQUATION. It now remains to ﬁnd the functional equation of L(s,ρ). It will beoftheform

Λ(s)= W (ρ) Λ(1 − s) with Λ(s)= C(s, ρ)LR(s, ρ) L(s, ρ) .

Here ρ is the representation dual to ρ, LR(s,ρ) is a product of Γ•factorsb overb all realb valuations of F , C(s,ρ)= M s/2 is some term determined by the ramification of E/F as well as of ρ, and W (ρ) is the analogue of the Gaussb sums that appear in the functional equations of Hecke’s L•functions. It has absolute value 1. It remains to define these terms explicitly, and to give a rough idea of why the functional equation holds. The basic fact in all cases is Brauer’s theorem, which asserts that any representation of a finite group is a virtual sum of representations induced from one•dimensional representations of subgroups. This goes together with properties (c) and (d) above. All four properties (a)–(d) are equally valid for the Γ, discriminant, and W (ρ) factors. (These are discussed in [Martinet:1979]. Dealing with the Artin conductor and W (ρ) is tricky.)

REAL VALUATIONS. Set −s/2 s ΓR(s)= π Γ(s/2), ΓC(s) = 2(π Γ(s) .

If v is a complex valuation of F , set dim ρ Lv =ΓC (s) .

Now suppose v to be a real valuation of F . Over it sit either complex or real valuations of E, with local Galois group either {1} or {1, σ}. Let Fv be the generator, in either case. Let ρ = ρ+ ⊕ ρ− be the eigenspace decomposition of ρ(Fv ). Then dim ρ+ dim ρ− Γv(s)=ΓR (s)·ΓR (s + 1) . Applications of CFT (9:09 p.m. October 5, 2013) 5

RAMIFICATION. To get this right, we need the notion of the conductor c of ρ, an ideal of Fv. I’ll just refer you to [Martinet:1979] and §§VI.1–3 of [Serre:1968]. It’s a very delicate and intriguing matter. At any rate, set

dim ρ M(ρ)= Nc·(NdF/Q) .

ROOT NUMBERS. The number W (ρ) appears as a consequence of applying Brauer’s theorem. Such expressions are not unique. If ρ is a one•dimensional representation, Artin reciprocity tells us that the L•funtion is one of Tate’s. In this case W (ρ) can be expressed as a product of local Gauss sums Wp(ρ,ψ). (Here ψ is a local additive character. I have not mentioned’it before, but it seems clear now that defining such local factors requitres such a choice, although in the final global product the choices don’t matter.) Brauer’s theorem can give several different expressions of ρ as a virtual sum of one•dimensional representations, and it is not at all clear that these don’t give rise to different local factors. Bernard Dwork showed that in fact one can find a well defined factorization for W 2(ρ), and Langlands completed this to find unique local factors Wv(ρ,ψ) in all cases. It is not at all a trivial matter. [Deligne:1972] gave a new proof of Langlands’ result. It used a few simple tricks about local Gauss sums together with a global argument. He was able to isolate local factors in global L•functions. [Tate:1979] gives a short and complete account of Deligne’s proof.

THE FUNCTIONAL EQUATION. AS does the functional equation. But because Bruaer;s theorem gives a virtual representation, we get something meromorphic, not even remotely seen to be holomorphic.

MORE ABOUT ROOT NUMBERS. Langlands’ key addition to Dwork’s work is a factor λ(E/F,ψ) for every p•adic extension E/F . The point is that property (c) (induction) for W (ρ) holds globally, it does not hold locally. It does hold for virtual representations of dimension 0, but otherwise one needs the λ•factor. The correct formula (induction from F to E) is is

W (ρ∗,ψ)= λ)E/F (ψ)W (ρ,ψ)) .

This is explained very clearly in [Tate:1979], and it is also worthwhile to read the introduction to [Langands:1967]. The numbers λE/F occur everywhere in Langlands’ work, for example in the theory of endoscopic transfer.

WHY THE ADDITIVE CHARACTERS. Tate’s thesis fixes additive characters of the fields Qp and R. This determines additive characters of extensions by composing with the trace. The important point is that these choices give rise to a character ψ of the adele` group AF that is trivial on F . But these choices are not canonical, × since ψ(ξx) is also satisfactory if ξ lies in F . Langlands allows differet choices of additive characters, and then chooses additive Haar measures on local fields so as to get the Fourier transform self•dual. Deligne allows arbitrary measures, and thus refers to W (ρ, ψ, dx). It is easy enough to move around all these conventions. iu s

EXTENDING TO THE WEIL GROUP. One can extend Artin’s L•functions to those of representations of local and global Weil groups. For the real valuations there are significant modifications, but they are not difficult to handle. For p•adic valuations, the important fact is that every finite dimensional representation of Wk is a twist of one of a Galois quotient by some |w|s. One usually overlooked feature of replacing Galois groups by Weil groups is that now we see where the s in − L•functions comes from, since ρ(F)Np s = ρ(F)|F|s.

REFERENCES

11. Pierre Deligne, ‘Les constantes des equations´ fonctionelles des fonctions L’, pp. 501–597 in Lecture Notes in Mathematics 349, Springer, 1972. The main result is a relatively simple proof of Langlands’ theorem about local root numbers. But this paper does not convey the true ﬂavour of the subject, which started out as a matter of proving equalities of products of Gauss sums. 12. Albrecht Frohlich¨ (editor), Algebraic number ﬁelds: L•functions and Galois properties, proceedings of a symposium in Durham, Academic Press, 1977. 13. Robert P. Langlands, notes on the functional equation of the Artin L•functions, available from Applications of CFT (9:09 p.m. October 5, 2013) 6

http://http://publications.ias.edu/rpl/section/22 For better or worse never completed. Langlands’ comments on his web site, especially those about Dwork’s role, are valuable. The introduction to he notes is also instructive. 14. Jacques Martinet, ‘Character theory and Artin L•functions’, pp. 1–87 in [Frohlich:1977].¨ 15. Peter Roquette, ‘On the history of Artin’s L•functions and conductors’, available at http://www.rzuser.uni-heidelberg.de/ ci3/lfunktio.pdf 16. Jean•Pierre Serre, ‘Sur la rationalite´ des representations d’Artin’, Annals of Mathematics 72 (1960), 405–420. 17. ——, Linear representations of ﬁnite groups, Sptringer, 1977. English translation of French original pub• lished by Hermann. Chapter 10 proves Brauer’s theorem. 18. John Tate, ‘Local constants’, pp. 89–131 in [Frohlich:1977].¨ This and [Martinet:1997] are readable and efﬁcient. Tate includes a complete account of Deligne’s proof of Langlands’ theorem of the existence of local root numbers, and a good treatment of the λ•factors.

5. Langlands’ classiﬁcation for tori

Langlands has conjectured that admissible representations of the group of rational points on reductive groups over local fields are more or less parametrized by admissible homomorphisms from Weil groups into L•groups. He verified this for tori. This case is a somewhat straightforward generalization of local class field theory—the case of split tori is just a restatement of the reciprocity isomorphism.

REFERENCES

19. Jeffrey Adams and David Vogan, ‘L•groups, projective representations, and the Langlands classification’, American Journal of Mathematics 114 (1992), 45–138. A potentially good account for real tori, but distressingly verbose. 20. Clifton Cunningham and Monica Nevins (editors), Ottawa lectures on admissible representations of p•adic groups, AMS, 2009. 21. Robert Langlands, ‘Representations of abelian algebraic groups’, Pacific Journal of Mathematics (special issue dedicated to Olga Taussky Todd) (1997), 231–250. Not easy to read. More than most of Langlands’ papers, this is largely written with himself in mind. But parts, especially the introduction, are very valuable. 22. ——, ‘On the classification of representations of real reductive groups’, preprint, 1974. Pages 19–20 prove the classification of characters of real tori in terms of the L•group. 23. James Milne, Arithmetic duality theorems, available at http://www.jmilne.org/math/Books/adt.html Look at Section 8. Maybe the most readable account. 24. J•K. Yu, ‘Local Langlands correspondence for tori’, Chapter 7 in [Cunningham•Nevins:2009]. The proofs in other places use all of the mechanism of class field theory to prove the main result, whereas Yu gives a short account based on simple and natural axioms. This works only for local fields—there are good reasons why nothing so simple works for global ones. He supplements the basic theorem with some remarks on how characters of tori relate to natural filtrations on Wk and T (k). This is reminiscent of Hasse’s results for the reciprocity isomorphism (Chapter XV of [Serre:1968].) But as far as I can tell, questions about wild ramification are completely open.