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. Emil Artin 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 reciprocity law 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 character.” 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 ideal 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 Dedekind zeta function 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 Hecke character 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. Representation theory 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 Galois extension 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) ) .