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Home , Dedekind zeta function, Hecke character, Langlands program, Ramification group, Ray class field

The Art of Artin L-Functions

Minh-Tam Trinh 15 October 2015

An apocryphal story tells us that Mozart composed the overture to Don Giovanni on the night before/morning of the premiere (29 Oct. 1787, in fact). But it’s a very good overture. . .

1

We all know that the theory of L-functions begins with Euler’s “observation”: ∞ X Y 1 (1.1) n−1 = . 1 − p−1 n=1 p That is to say, the partial sums of the left-hand side approach the partial products of the right-hand side asymptotically. For both Euler, as well as Riemann in his famous 1859 memoir, the left-hand side, or additive description, was the starting object and the right-hand side, or multiplicative description, was a deduction. Question. What was the first kind of L-function to be defined a priori as a product, not as a series? Answer. Artin L-functions. lived from 1898 to 1962. He received his doctorate in 1921 and married Natascha Jasny in 1929. From 1922 to 1937, he held academic positions in Hamburg, leaving only because the rise to power of the Nazis threatened his wife, who was Jewish, and his cultural and moral values at large. Of the four great papers listed below, I will try mainly to explain the history of ideas that inspired the two in bold: 1. 1923, “Über ein neue Art von L-Reihen.” Gave the first definition of an Artin L-function and established its functorial properties. 2. 1927, “Beweis des allegemeinen Reciprozitätgesetzes.” Established the of class field theory. 3. 1930, “Zur Theorie der L-Reihen mit allegemeinen Gruppencharakteren.” Re- vised the 1923 definition to properly incorporate the places of ramification. 4. 1931, “Die gruppentheoretische Struktur der Diskriminante algebraische Zahlkörper.” De- scribed Artin conductors and their properties. Another paper relevant to us will be Richard Brauer’s 1947 “On Artin’s L-series with general group .” 2

The timeline of L-functions looks like this: P Q 1. 1737, Euler, “Various observations about infinite series.” Introduced n = p. 2. 1837-1840, Dirichlet. 3. 1859, Riemann, “On the number of primes less than a given magnitude.” Introduced functional equation and meromorphic continuation. 4. 1863, Dedekind. 5. 1918-1920, Hecke. In more detail: Initially, there were only the Riemann zeta ζ and Dirichlet L-functions L(s, χ). Kummer and Dedekind intuited that the former of these had analogues in other number fields K: Hence they introduced

X −s (2.1) ζK (s) = N(a) . aCOK Weber similarly generalized L(s, χ) by allowing χ to be any finite-order ray-class character. To explain this remark, observe that the naive generalization of Z/nZ is OK /a. But whereas you can lift naturally from Z/nZ to Z, you cannot do the same from OK /a to OK because the latter does not have upf of elements. Nonetheless we do have upf of ideals, as Dedekind showed. So Weber instead lifted to characters of groups of classes. Recall that a modulus of K is a formal product m = m0m∞, where m0 C OK is a nonzero ideal and m∞ is a subset of the infinite places. We define Im to be the group of nonzero fractional ideals generated by the integral ideals that do not divide m. Notice that if n | m, meaning m ⊆ n, then there is an inclusion In ,→ Im. We define Pm to be the subgroup generated by the principal fractional ideals for which we can choose a generator α that satisfies the following conditions:

1. ordp(α − 1) ≥ ordp(m0).

2. τ(α) > 0 for all τ ∈ m∞.

The ray class group of modulus m is Cm = Im/Pm.

Example 2.1. Take K = Q and m = n. Then Pm is generated by the ideals that can be written aZ for some a such that a ≡ ±1 (mod n), so we get

× (2.2) Cn ' (Z/nZ) /h±1i.

But if m = n∞, then Pm is only generated by the ideals aZ such that a ≡ 1 (mod n), so we get

× (2.3) Cn∞ ' (Z/nZ) . Weber and Hecke seem to have understood(?) that, for each modulus m, there exists a finite abelian extension K(m)/K ramified only at the places dividing the modulus.1 We say that K(m)

1Explain what ramification at infinite places means?

2 is the ray class field of K of modulus m. For example, K1 is called the Hilbert class field of K. With the hindsight of the work of Artin-Takagi, we now know that there is an isomorphism

(2.4) Cm ' Gal(K(m)/K) and that every finite abelian extension of K is contained in some ray class field, meaning the ray class fields are sufficient to characterize the explicit class field theory of K. At any rate, Weber introduced:

X −s (2.5) L(s, χ) = χ(a)N(a) aCOK for any character χ of Im that is trivial on Pm, i.e., any character of Cm. We can now state the lovely theorem that relates Dedekind zeta functions to Weber L-functions: Theorem 2.2 (Hecke-Weber). Let K be a number field, and let m be a modulus for K. If L/K is a subextension of K(m)/K, and H is the corresponding normal subgroup of Cm, then Y Y (2.6) ζL(s) = L(s, χ) = ζK (s) L(s, χ)

χ6=χ0 χ∈C\m/H formally, with convergence in the half-plane Re s > 1. In particular, ζL/ζK is entire. Remark 2.3. Hecke’s contributions seem to have been showing that 1) the Galois correspondence could be transferred to the ideal-class side and 2) the L(s, χ) for nonprincipal χ are entire.

Example 2.4. Take K = Q and L = Q(i). Then Gal(L/K) ' (Z/4Z)× and L is in fact the ray class field for 4∞. The Hecke-Weber factorization is Y 1 (2.7) ζ (s) = Q(i) 1 − (p)−s p N 1 Y 1 Y 1 = 1 − 2−s (1 − p−s)2 1 − p−2s p splits p is inert Y 1 Y 1 Y 1 = 1 − p−s 1 − p−s 1 + p−s p p≡1 (mod 4) p≡3 (mod 4) = ζ(s)L(s, χ), where χ is the sign character of (Z/4Z)×. We can deduce the functional equation and analytic properties of the from those of the Weber L-functions. More precisely, Hecke showed that the latter can be completed to functions Λ(s, χ) with meromorphic continuation to the plane, such that

(2.8) Λ(s, χ) = ε(s, χ)Λ(1 − s, χ), where |ε(s, χ)| = 1. He also introduced a further refinement of Weber’s characters, namely, the 2 Grössencharakteren or Hecke characters. Observe that, as Im/Pm is finite, all Weber characters are a fortiori of finite order. Hecke characters, by contrast, can be of infinite order.

2Beyond the fact of the language itself, there is something ineffably German about this nomenclature.

3 At this point, I was going to try to explain formally the historical, ideal-theoretic definition of Hecke characters and relate it to our modern, idele-theoretic one via concrete examples, but it’s rather laborious and isn’t directly relevant to Artin. So I shall relegate such discussion to an appendix. But let me recall that, idelically speaking, a of K is a character of × × AK that factors through K .

3

What follows is based heavily on Cogdell’s exposition “On Artin L-functions.” In Hamburg, Artin’s intellectual passion was class field theory. He wanted to extend the Hecke- Weber factorization, describing Dedekind zeta functions in abelian extensions, to a nonabelian formulation. He was particularly inspired by the work of Frobenius in two areas: 1. of finite groups. 2. Actions of Galois groups on primes in number fields. His idea was to attach L-functions to representations of Galois groups. A first motivation was the form of the Euler factors in a Hecke L-function. × × Let K be a number field, and let χ : AK → C be a (unitary) Hecke character. If p is a nonzero prime of K, then the Euler factor of L(s, χ) at p is

−s −1 (3.1) Lp(s, χ) = (1 − χ($p)(Np) ) ,

3 where above, $p is an arbitrary uniformizer. Taking the logarithm,

∞ X ( p)−ms (3.2) log L (s, χ) = − χ($ )m N . p p m m=1

If we want to replace χ with a representation V of Gal(L/K) for some finite L/K, no longer necessarily 1-dimensional, then it is natural to replace χ($p) with χV (σP), where σP is a Frobenius lift at p. The two potential obstacles are:

1. If L/K is not abelian, then there are multiple decomposition groups DP|p. 2. If p ramifies in L, then the Frobenius lift to a given decomposition group is a coset, not just an element. Obstacle (1) is resolved because the decomposition groups are all conjugate, so the trace of Frobenius is independent of the choice. In particular, we can write σp in place of σP without loss of information. In his 1923 paper, Artin dealt with obstacle (2) by simply ignoring the primes dividing the discriminant of L/K. Altogether, to a representation V of a finite Galois extension L/K, Artin attaches an L-function L(s, L/K, V ), whose Euler factor at an unramified prime p of K is

−s −1 (3.3) Lp(s, L/K, V ) = det(1 − χV (σp)(Np) ) .

3 × The result is independent of the choice of $p because χ factors through K ?

4 In words, Lp(s, L/K, V ) is the inverse of the characteristic polynomial of σp under V , evaluated −s Q at (Np) . Standard arguments show that L(s, L/K, V ) = p Lp(s, L/K, V ) converges in the half-plane Re s > 1. I will here break with chronology and explain the hack that fixes the definition for the ramified primes. The key is to observe that, if p is unramified, then χV (σp) depends only on ResGal(L/K) V . But the inertia group I is normal in D , so we can everywhere replace Dp p p Gal(L/K) Gal(L/K) D Res V with Res Inf p V Ip . That is, the definition of the Euler factor of Dp Dp Dp/Ip L(s, L/K, V ) at an arbitrary prime p should be

−s −1 (3.4) Lp(s, L/K, V ) = det(1 − χV |V Ip (σp)(Np) ) , as Artin announced in 1930.

3.1 Functoriality Throughout the rest of this exposition, L/K is a finite Galois extension of number fields and G = Gal(L/K). All representations will be finite-dimensional and over C. When I write §,I refer to a section in Artin’s 1923 paper. Having defined his new type of L-functions, Artin’s initial goal was to generalize the Hecke- Weber result. To this end, he made the skillful observation that Y (3.5) ζL(s) = L(s, χ) χ was analogous to the decomposition

M ⊕ dim ρ (3.6) rG = ρ , ρ∈Gˆ of the rG of G into irreducibles. Conjecturing an analogous factorization for L(s, L/K, V ), he is led to (§2-3) the following properties:

1. Additivity. If V1,V2 are representations of G, then

(3.7) L(s, L/K, V1 ⊕ V2) = L(s, L/K, V1)L(s, L/K, V2).

This would explain how to translate the additive structure of (3.6) into the multiplicative structure of (3.5).

2. Inflation. If Q is a quotient group of G, corresponding to a subextension E/K, and V is a representation of Q, then

G (3.8) L(s, L/K, InfQ(V )) = L(s, E/K, V ).

The motivation is the requirement that L(s, L/K, 1L/K ) = ζK (s), where 1L/K is the trivial representation of G and ζK (s) = L(s, K/K, 1K/K ).

5 3. Induction. If H is a subgroup of G, corresponding to a subextension L/E, and W is a representation of H, then

G (3.9) L(s, L/K, IndH (W )) = L(s, L/E, W ).

The motivation is the requirement that L(s, L/K, rG) = ζL(s) = L(s, L/L, 1L/L), observing G that rG = Ind1 1L/L. Indeed, once all of these properties are proven, the following factorization is a corollary (§4) of (3.6): Theorem 3.1 (Artin). Let L/K be an arbitrary finite Galois extension of number fields. Then

Y dim V (3.10) ζL(s) = ζK (s) L(s, L/K, V ) V ∈Gˆ V 6=1L/K formally, and convergence holds in the half-plane Re s > 1.

3.2 Reciprocity From comparing this result with Thm. 2.2, it is natural to ask whether there is a bijection between, say, the characters of Gal(K(m)/K) and the characters of Cm for a fixed modulus m. And it was this question that provoked Artin to formulate the statement of Artin Reciprocity in his 1923 paper, and ultimately, to prove it in 1927. The modern version is: Theorem 3.2 (Reciprocity). Suppose L/K is a finite Galois extension of number fields, of abelian . Then the map

× × × (3.11) AK /(K NL/K (AL )) → Gal(L/K) that sends (the image in the left-hand side of) p to the Frobenius lift σp is a group isomorphism functorial in L and K. For all P | p, it specializes to an isomorphism

× × (3.12) Kp /NLP/Kp (Lp ) → DP|p

e e e that restricts to an isomorphism 1 + p → DP|p for all e ≥ 0, where D denotes the eth ramification group. In the 1923 paper, Artin assumes the reciprocity law to hold true and deduces (§5) the compatibility of his L-functions with Hecke’s. (Up to the factors at primes of ramification, though this issue vanishes with the correct definition of the ramified Euler factors.) Corollary 3.3. Suppose L/K is a finite Galois extension of abelian Galois group. Let χ : Gal(L/K) → C× be an irreducible character, and let χ˜ be the associated Hecke character under the reciprocity law. Then L(s, L/K, χ) = L(s, χ˜). Proof. The point is that the Euler factor of L(s, L/K, χ) (resp., L(s, χ˜) at p “collapses” if and only if χ (resp., χ˜ is ramified at p, meaning χ|I = 1 (resp., χ˜| × = 1. And χ ramifies if and p Op only if χ˜ ramifies, due to the isomorphism in the reciprocity law.

6 3.3 Complements Before discussing §6 of the 1923 paper, let me quickly survey §7-9, the remaining sections. In §7 he states a generalization of Dirichlet’s Theorem on primes in arithmetic progressions. It looks like a version of Chebotarev?

Theorem 3.4. Let L be an arbitrary number field. Let C be a fixed conjugacy class of Gal(L/Q), and let πC(X) be the number of primes p of L with Np ≤ X. Then Z X dt −A(log X)1/2 (3.13) πC(X) = C + O(Xe ), 2 log t where C, A > 0 are constants with 1/C = P χ(σ)χ(τ) for any σ, τ ∈ C. χ∈Gal(\L/Q) Artin says the above is proved from considering the logarithmic derivative of the L(s, L/Q, χ)’s, applying “well-known methods,” then summing over χ. In §8 he proves that there are no “multiplicative relations” among the L-functions attached to irreducible representations. That is,

Theorem 3.5. Let L/Q be a finite Galois extension. If

Y a (3.14) L(s, L/Q, χ) χ = 1

χ∈Gal(\L/Q for some aχ ∈ Q, then aχ = 0 for all χ. The proof is to take logarithmic derivatives of both sides, then invoke the preceding prime- P density theorem to deduce that χ aχχ(σ) = 0 for all σ ∈ Gal(L/Q). Then by Dedekind’s lemma on independence of characters(?), we conclude. In §9, he considers an icosahedral field L over Q, i.e., one for which Gal(L/Q) ' A5, and decomposes the Dedekind zeta functions for the various intermediate fields in terms of the and the L-functions of other irreducible representations of Gal(L/Q).

4

In §6, Artin’s aim is the functional equation of his L-functions. To do this, he considers a different approach to generalizing the Hecke-Weber factorization, one perhaps more natural from the hindsight of the . That is, instead of factoring a Dedekind zeta function into Artin L-functions, can we factor an Artin L-function into Hecke L-functions? The reason this question is reasonable in the first place is the induction property. In representation-theoretic terms, we are asking whether the character of the regular representation of a finite group G can be written as a linear combination, with nonnegative integer coefficients, of characters induced from abelian subgroups of G. (More generally, a character is called monomial iff it is induced from a 1-dimensional character of a subgroup.) So in a paper about , Artin ends up proving a minor masterpiece of pure representation theory:

Theorem 4.1 (Artin). If G is any finite group, then every character of G is a Q-linear combination of characters induced from cyclic subgroups.

7 But since the coefficients are merely rational numbers, not nonnegative integral integers, in the above theorem, we do not actually obtain a factorization of our L-functions as products of Hecke L-functions with integer multiplicities. In 1947, Brauer improved the situation, as follows. If p is a prime, then we say that an element h of a group H is p-regular iff its order is coprime to p. We say that H is p-elementary iff it is the direct product of a p-group with the cyclic subgroup generated by some p-regular element. Theorem 4.2 (Brauer). If G is any finite group, then every character of G is a Z-linear combination of characters induced from elementary subgroups. Corollary 4.3. Every Artin L-function factors as a ratio of products of Hecke L-functions. Proof. By Brauer induction and the induction property for Artin L-functions, we at least get a ratio of products of Artin L-functions attached to 1-dimensional representations, possibly for smaller subextensions. But any such representation factors through the quotient of the Galois group associated to the maximal abelian subextension of that particular subextension. So by the inflation property of Artin L-functions and Cor. 3.3, we’re done. Corollary 4.4. Every Artin L-function admits a meromorphic continuation to the plane and a functional equation with symmetry across Re s = 1/2. Proof. The conclusion holds for Hecke L-functions, by works of Hecke, Iwasawa, and Tate.4 Just as Hecke L-functions for nonprincipal characters are holomorphic, one can ask if Artin L-functions of nonprincipal irreducible characters are holomorphic. Apparently, this would be resolved by the Langlands Program: I would be interested to hear an explanation if anyone knows one. Conjecture 4.5 (Artin). If L/K is a finite Galois extension and V is a nontrivial irreducible representation of Gal(L/K), then L(s, L/K, V ) is entire.

4.1 Explicit Brauer Induction

It is not hard to construct an S5-extension of Q as a splitting field of some f. Indeed, recall that Sp for prime p is generated by any combination of transposition and p-cycle. If we take f to be of degree p and to have exactly 2 nonreal roots, then we get the transposition from the conjugation of the nonreal roots, and the p-cycle from Cauchy’s Theorem and the fact that [Q(a): Q] = p for any of the roots α of f. Setting (4.1) f(X) = X5 − 4X − 1, graphing f (or using calculus) shows that f has exactly 2 nonreal roots, as required. Example 4.6. I rushed to copy/paste the following example from an earlier exposition I had written, so I may be foggy on the details. Let V be the standard representation of S5, a represntation of dimension 4. For each cycle type c in S5, let σc be an element of that cycle type. The class function χV is given by the following table: 1 σ σ σ σ σ σ (4.2) 2 2,2 3 3,2 4 5 χV 4 2 0 1 −1 0 −1

4In particular, Tate’s thesis.

8 × 2πi/#hσci S5 Let ψc : hσci → be the character that sends σ 7→ e , and let χc = Ind (ψc). Using C hσci the Mackey formula

1 X −1 (4.3) χc(g) = ψc(x gx) #hσci x∈S5 −1 x gx∈hσci for induced characters, we have:

1 σ2 σ2,2 σ3 σ3,2 σ4 σ5 (4.4) χ3,2 20 −2 −1 1 χ5 24 −1

4 Therefore, χV = χ5 − χ3,2. It follows that if L is the splitting field of X − 4X − 1 over Q and we identify Gal(L/Q) with S5, then

L(s, ψ5) (4.5) L(s, χV ) = L(s, ψ3,2) in the notation above.

A

The following discussion is indebted to Jerry Shurman’s writeup on Hecke characters. Let K be a number field, and let f be an arbitrary nonzero of K. Recall × that we write ∞K for the group of the product of the archimedean completions of K and × × identify K with its diagonal image in ∞K . Let Kf be the group of nonzero principal fractional ideals of K coprime to f, and let

× × × (A.1) Kf,1 = (1 + fKf ) ∩ K .

× × Note that Kf,1 is all of 1 + fKf if and only if f is not OK itself. For integral m C OK , we have a natural isomorphism

× × × (OK /m) → Km /Km,1 (A.2) × x + m 7→ xKm,1

× × Injectivity holds because Km,1 ∩ OK = 1 + m. Surjectivity holds because for all xKm,1, we can × × × × find y ∈ OK such that xy ∈ OK but also xyKm,1 = xKm,1 (by Chinese Remainder?). Remark A.1. As further motivation, observe that the local version of the isomorphism above is

(A.3) O×/(1 + pe) ' lim(O /pe)× p ←− p e Intuitively, for integral f, a Hecke character of conductor f will be a continuous group × × × morphism If → C that induces a morphism (OK /f) → C upon incorporating the needed data at archimedean places. The last piece of information we need is, an ∞-type for K is a

9 × × continuous character ∞K → C . Then formally, a Hecke character of K of conductor f and 5 ∞-type χ∞ is a (continuous) character

× (A.4) χ : If → C

−1 × such that χ(xOK ) = χ∞ (x) for all x ∈ Kf,1. × × × × × In particular, the (OK /f) -type of χ is the character  :(OK /f) ' Kf /Kf,1 → C defined by

(A.5) (x + f) = χ(xOK )x∞(x)

0 We say χ is imprimitive iff it factors through If0 for some strictly coarser modulus f | f. It is primitive otherwise. In the idelic version, this problem of imprimitivity does not arise.

× Example A.2. Recall that Z[i] is a PID. Let χn : IQ(i) → C be the continuous group morphism

4n (A.6) χn(xZ[i]) = (x/|x|)

4 × Then χn is well-defined because u = 1 for every unit u ∈ Z[i] . Its conductor is Z[i], so its −1 -type is trivial and its ∞-type is χ∞(x) = χn(xZ[i]) .

5This treatment differs from the usual practice of subsuming the conductor within the modulus of χ, which includes both data at infinite as well as finite places. However, Shurman’s definition using ∞-types is clearer and more appealing to me.

10 References

[Ar1] E. Artin. “Uber eine neue Art von L-Reihen.” Abh. Math. Sem. Hamburgischen Univ., Vol. 3, No. 1 (1923), 89-108. [Ar2] E. Artin. “Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren.” Abh. Math. Sem. Hamburgischen Univ., Vol. 8, No. 1 (1930), 292-306.

[Br] R. Brauer. “On Artin’s L-Series with General Group Characters.” Annals of , Vol. 48, No. 2 (Apr. 1947), 502-514. [BT] R. Brauer & J. Tate. “On the Characters of Finite Groups.” Annals of Mathematics, Vol. 62, No. 1 (Jul. 1955), 1-7. [Ca] J. W. S. Cassels. “Global Fields.” . Ed. J. W. S. Cassels & A. Fröhlich (1967), 42-84.

[Co] K. Conrad. “Galois Theory at Work.” (2013). http://www.math.uconn.edu/ [He1] E. Hecke. “Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, I.” Mathematische Zeitschrift (1918), 357-376.

[He2] E. Hecke. “Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, II.” Mathematische Zeitschrift (1919), 11-51. [He3] E. Hecke. “Über die Zetafunktion beliebiger Zahlkörper.” Nachrichten der K. Gesellschaft der Wissenschaftem zu Göttingen, Mathematisch-physikalische Klasse (1917), 299-318.

[Ho] A. Holmström. “Harmonic Analysis on Number Fields.” Royal Institute of Technology. Stockholm, Sweden (2005). [Iw] K. Iwasawa. “A Note on Functions.” Proceedings of the International Congress of Math- ematicians, Vol 1 (1950), 322. [MM] M. R. Murty & V. K. Murty. Nonvanishing of L-Functions and Applications. Birkhauser Verlag (1997). [Ne] J. Neukirch. Algebraic Number Theory. Springer (1999). [Se] J.-P. Serre. Linear Representations of Finite Groups. Trans. L. L. Scott. Springer-Verlag (1977).

[Sn] V. P. Snaith. Explicit Brauer Induction. Cambridge University Press (1994). [Ta] J. Tate. "Fourier Analysis in Number Fields and Hecke’s Zeta-Functions (Thesis, 1950)." Algebraic Number Theory. Ed. J. W. S. Cassels & A. Fröhlich (1967), 305-347.

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