Emil Artin knew that given a finite extension of $L/\mathbb{Q}$, the local factor of the zeta function $\zeta_{L/\mathbb{Q}}$ at the prime $p$ should be $\displaystyle\prod_{\mathfrak{p}|p}\frac{1}{1 – N(\mathfrak{p})^{-s}}$. He also knew that if $L/K$ is a class field then $\displaystyle\prod_{\mathfrak{P}|\mathfrak{p}}\frac{1}{1 – N(\mathfrak{P})^{-s}} = \displaystyle\prod_{\chi}\frac{1}{1 – \chi{(Frob{_\mathfrak{p})}}\cdot N(\mathfrak{p})^{-s}}$ where $\mathfrak{P}$ runs over all primes in $L$ lying above $\mathfrak{p}$ and $\chi$ runs over all characters of $Gal(L/K)$.
It's natural then to
- Define $L$-series attached to characters on $Gal(L/K)$.
- Recognize that the definition makes sense whether or not $L/K$ is a class field.
- In light of the fact that characters are $1$-dimensional representations of $Gal(L/K)$, ask whether there's a good definition of the $L$-series attached to a higher dimensional representation of non-abelian $Gal(L/K)$.
But having come this far, how does one then arrive at the definition of the local factor of an $L$-series attached to a representation $\rho: Gal(L/K) \to GL_{n}(\mathbb{C})$ at a prime $\mathfrak{p}$ unramified in $K$ as
$\displaystyle \frac{1}{\det(Id – \rho(Frob_\mathfrak{p})N(\mathfrak{p})^{-s})}$
?
To be sure
- It specializes to the definition of the $L$-series attached to a character on $Gal(L/K)$.
- It's well-defined (independent of which member of the conjugacy class $Frob_\mathfrak{p}$ one chooses).
- One has the theorem $\zeta_{L/\mathbb{Q}} = \prod_{\rho} L(\rho, s)$ where $\rho$ ranges over irreducible representations of $Gal(L/\mathbb{Q})$, generalizing the analogous fact for characters on Galois groups of class fields.
And perhaps the three properties listed above are sufficient to uniquely determine the definition. (Maybe one needs more than the above three, I would have to think about it it.) Maybe this is how Artin discovered the definition. This line of thinking is similar to Feynman's heuristic derivation of Heron's formula. But I somehow feel as though this doesn't get at the essence of things. Is there a way of thinking about the definition of an Artin L-series that gives it more of a sense of inevitability and canonicity?
[Reposted from mathstackexchange.]
Best Answer
Artin's work on zeta functions began in 1923 (actually zeta functions had already played a role in his thesis on quadratic extensions of the rational function field) with an article "On the zeta functions of certain algebraic number fields". There he studied a problem due to Dedekind which asked whether the zeta function of a number field is always divisible (in the sense that the quotient is entire) by the zeta function of any of its subfields. Dedekind had proved this for purely cubic fields, and for abelian (and in fact metabelian, then called metacyclic) extensions it follows from the decomposition of Dedekind's zeta functions into a product of abelian L-series due to Takagi's class field theory.
Artin then computed explicitly the zeta functions for subfields of an $S_4$-extension, where the factors contributed by a prime ideal ${\mathfrak p}$ depends on the decomposition group of ${\mathfrak p}$, and then he sketched a similar calculation for the icosahedral group. For unramified primes, these factors all have a natural interpretation in terms of the Frobenius automorphisms, or, in other words, come from a Galois representation. One can do worse than read Harold Stark's beautiful article in the book "From number theory to physics", where even simpler examples all presented in all their glory.
In Artin's first article on L-series (On a new kind of L-series, 1923) Artin defined the Euler factors of the L-series attached to a Galois representation only for unramified primes. This was sufficient (if not very satisfying) for the following reason: Artin could write the zeta function of $K$ and all of its subfields as products of his L-series. Hecke had shown in 1917 that L-series whose Euler factors agree up to at most finitely many primes actually have equal Euler factors if both L-series satisfy the same functional equation. So if you can show that Artin's L-series satisfy a functional equation with suitably defined (but not explicitly known) factors at the ramified and infinite primes, then everything is fine. At the end of this artice, Artin takes up his example of the icosahedral group again.
In his sequel "On the theory of L-series with general group characters" from 1930, Artin observed that the state of the theory was not satisfactory and proceeded to define the "local" factors (local class field theory was being developed simultaneously by Hasse; Artin's reciprocity law had allowed a new approach to the norm residue symbols, and this led more or less automatically to local class field theory) from the start. He does this by starting with a Galois representation, observing that for ramified primes, the "Frobenius automorphism" is only defined up to elements from the inertia group $T$, and then constructs a representation of $Z/T$, the factor group of decomposition modulo inertia group; then he uses this "piece" of the representation for defining the local factors at ramified primes. Parts of the necessary arguments can be found in Artin's article on the group theoretic structure of the discriminant in algebraic number fields that appeared in print in 1931.
In his letter to Hasse from Sept. 18, 1930, Artin gives the following explanation (the notation is essentially the same as in his articles):
Let ${\mathfrak p}$ be a prime ideal, $\sigma$ the associated substitution in $K/k$, which is not uniquely determined, ${\mathfrak T}$ the inertia group, and $e$ its order. Set $$ \chi({\mathfrak p}^\nu) = \frac{1}{e} \sum_{\tau \in {\mathfrak T}} \chi(\sigma^\nu\tau)\ , $$ which is the mean of all possible values. Then $$ \log L(s,\chi) = \sum_{{\mathfrak p},\nu} \frac{\chi({\mathfrak p}^\nu)}{\nu N{\mathfrak p}^{\nu s}} $$ is the complete definition also for divisors of the discriminant. $L(s,\chi)$ can be written as usual as a product of the form $$ L(x,\chi) = \prod_{\mathfrak p} \frac{1}{|E-N{\mathfrak p}^{-s} A_{\mathfrak p}|}, $$ where $A_{\mathfrak p}$ is a certain matrix attached to ${\mathfrak p}$ (which may be $0$) and only has roots of units as characteristic roots.
This explains the naive idea behind the definition: since the Frobenius is not well defined, take the mean over all possible values. Finally, Noah Snyder has written a very nice thesis on Artin L-functions, which contains a translation of Artin's 1923 article on L-series.