A natural way of thinking of the definition of an Artin $L$-function?

Question

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

1. Define $L$-series attached to characters on $Gal(L/K)$.
2. Recognize that the definition makes sense whether or not $L/K$ is a class field.
3. 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

1. It specializes to the definition of the $L$-series attached to a character on $Gal(L/K)$.
2. It's well-defined (independent of which member of the conjugacy class $Frob_\mathfrak{p}$ one chooses).
3. 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.]

Answer 1

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):

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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.

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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.

Answer 2

(1) One possible way to do this is to start from the axiom that the local factor only depends on the local behavior of the Galois representation. Locally at an unramified prime, the Galois representation is thus a representation of $\hat{\mathbb Z}$, in other words a matrix. Since we don't have a canonical basis that this matrix acts on, and for the conjugacy-class-of-Frobenius reason you state, the only numbers that we have access to are the coefficients of the characteristic polynomial of Frobenius.

When all you've got are the coefficients of a polynomial, plugging something in to that polynomial seems like a pretty obvious step. If you normalize it so that you get the right answer for characters, you get the Artin L-function.

(2) A slightly more rigorous argument uses the axiom that $L(\rho_1)L(\rho_2)=L(\rho_1\oplus \rho_2)$. Since (semisimple) Galois representations are locally direct sums of characters, you get a unique definition.

(3) As Qiaochu suggests, you can use the Lefschetz zeta function as motivation. However, I can't think of an argument why that should be related to number theory without going through Weil's zeta function and the Weil conjectures. Since Artin discovered his zeta function before Weil made his conjectures, this is unsatisfying as a historical approach, though it is critical to a modern understanding of the notion and can even serve as a motivation for the Riemann zeta function!

Perhaps Qiaochu can think of a better reason to use the Lefschetz zeta function.

(4) The stupidest possible way to do this is just to try to generalize the formula directly as possible. The old formula was $I_n-\rho(Frob_p) N(p)^{-s}$ and $\rho(Frob_p)$ is still a matrix while $N(p)^{-s}$ is still a scalar, but everyone knows that the higher-dimensional generalization of a scalar is the corresponding scalar matrix. Then you're left with a bunch of matrices you want to multiply together. You could just directly multiply them but this is bad for two reasons. First, you would really like a nice simple complex analytic function, and second, as you pointed out the matrices are only defined up to conjugacy.

The obvious thing to do here is to take the determinant. In particular, it does not even matter whether you take the determinant before or after multiplying the matrices, even with the conjugation indeterminacy.

Of course you probably screw up and try some wrong things, like $1-{\rm Tr}(\rho(Frob_p) N(p)^{-s})$. However the wrong things, because they're terrible, don't have any nice properties, so it's pretty easy to discard them and settle on the correct one. This is probably a more accurate representation of the historical process.

Answer 3

Here is a deeply ahistorical approach. Let's begin with the following topological analogue of a Galois extension with Galois group $G$, namely a Galois cover $Y \to X$ of spaces with Galois group $G$. Let's consider what the cohomology of these things looks like (as a topological analogue of taking etale cohomology). The cohomology of $Y$ over $\mathbb{C}$ decomposes as

$$H^{\bullet}(Y, \mathbb{C}) \cong H^{\bullet}(X, \mathbb{C}[G]) \cong \bigoplus_V H^{\bullet}(X, V) \otimes V^{\ast}$$

which means the following. First, $Y$ determines a $G$-local system on $X$, and so for any representation of $G$ we get an associated local system of vector spaces. $\mathbb{C}[G]$ denotes the local system on $X$ associated to the regular representation of $G$, which then decomposes as $\bigoplus_V V \otimes V^{\ast}$ where $V$ runs over irreducible representations of $G$. Hence cohomology with local coefficients in $\mathbb{C}[G]$ decomposes as a direct sum as above, where $H^{\bullet}(X, V)$ denotes cohomology with coefficients in the local system on $X$ associated to $V$. $G$ acts on $Y$ by covering transformations, and the corresponding action on the RHS is on the factors $\otimes V^{\ast}$. I may be off by a dualization here.

I claim this is a precise topological analogue of the factorization of the Dedekind zeta function of $L$ as a product of Artin L-functions. The idea is to think of the Dedekind zeta function as something determined by the etale cohomology of $\text{Spec } \mathcal{O}_L$, but multiplicatively, so that direct sums get sent to products, and then to hope / expect that the etale cohomology has a decomposition as above.

The construction which takes as input the etale cohomology of $\text{Spec } \mathcal{O}_L$ and returns as output the Dedekind zeta function has something to do with taking traces of Frobenius on the total symmetric power

$$\text{Sym}^{\bullet}(H^{\bullet}(\text{Spec } \mathcal{O}_L))$$

of the etale cohomology. At this point let me record the following useful observation from linear algebra.

Observation: Let $V$ be a finite-dimensional vector space on which a linear operator $A$ acts. Let $\text{Sym}^k(A)$ denote the induced action of $A$ on $\text{Sym}^k(V)$. Then

$$\sum t^k \text{ tr Sym}^k(A) = \frac{1}{\det (1 - At)}.$$

So, taking for granted that the etale cohomology at least heuristically decomposes into pieces $H^{\bullet}(\text{Spec } \mathbb{Z}, V)$, the corresponding thing to do to each piece is to take the trace of Frobenius on the total symmetric power of etale cohomology with local coefficients

$$\text{Sym}^{\bullet}(H^{\bullet}(\text{Spec } \mathbb{Z}, V))$$

and this is why one shouldn't be surprised to see the characteristic polynomial of Frobenius acting on $V$ in the local factors. (Here the topological analogue of the action of Frobenius only being well-defined up to conjugation is the holonomy around a loop only being well-defined up to conjugation.)

Answer 4

As @anon noted, it is very important to have the formation of these L-functions be compatible with induction (inducing repns...). A notion of induced repn was indeed available since the time of Frobenius et alia, and the proof of meromorphy (Artin and Brauer) used exactly that idea. In contrast, as @Filippo A.E. noted, an appreciation of "local computations" was less available at the time.

Thus, the "definition" of Artin L-functions was completely determined by compatibility with classfield theory and with induction (and certainly more-than-completely so after the Brauer theorem and its application here).

That viewpoint, with local-global ideas adjoined, and a bit more, was what Weil used for his extended class of L-functions.

Also, I think in the early 20th century people thought quite a lot about assembling abelian extensions into non-abelian towers to try to divine what "non-abelian classfield theory" should be, so the already decades-old Frobenius notion of "induction" would have been in play.

Answer 5

Since this is near the top again, I'll add what seems natural to me. Let $F/\mathbb{Q}$ be an $S_3$ extension, and let $C$ and $K$ be the cubic and quadratic subfields. Then the $\zeta$ functions of these number fields have Euler products.

Let $p \in \mathbb{Z}$ be a prime which is not ramified in $F$. Then the corresponding Euler factor in each of the fields $\mathbb{Q}$, $K$, $C$ and $F$ is dictated by whether the Frobenius element has conjugacy class $e$, $(12)$ or $(123)$. Here is the formula for the Euler factor in each case:

$$\begin{array}{|c|c|c|c|} \hline & e & (12) & (123) \\ \hline \zeta_\mathbb{Q} & (1-p^{-s})^{-1} & (1-p^{-s})^{-1} & (1-p^{-s})^{-1} \\ \hline \zeta_K & (1-p^{-s})^{-2} & (1-p^{-s})^{-1}(1+p^{-s})^{-1}&(1-p^{-s})^{-2} \\ \hline \zeta_C & (1-p^{-s})^{-3} & (1-p^{-s})^{-2} (1+p^{-s})^{-1}&(1-p^{-s})^{-1}(1+p^{-s}+p^{-2s})^{-1} \\ \hline \zeta_F & (1-p^{-s})^{-6} &(1-p^{-s})^{-3}(1+p^{-s})^{-3}& (1-p^{-s})^{-2}(1+p^{-s}+p^{-2s})^{-2}\\ \hline \end{array}$$

Looking at this table makes it natural to imagine additional rows for $L$-functions $L_1$ and $L_2$ with $\zeta_K = \zeta_{\mathbb{Q}} L_1$, $\zeta_C = \zeta_{\mathbb{Q}} L_2$ and $\zeta_F = \zeta_{\mathbb{Q}} L_1 L_2^2$ and Euler factors $$\begin{array}{|c|c|c|c|} \hline & e & (12) & (123) \\ \hline L_1 & (1-p^{-s})^{-1} &(1+p^{-s})^{-1} &(1-p^{-s})^{-1} \\ \hline L_2 &(1-p^{-s})^{-2} &(1-p^{-s})^{-1} (1+p^{-s})^{-1}& (1+p^{-s}+p^{-2s})^{-1}\\ \hline \end{array}$$ (It especially helps that Hecke and Dirichlet already studied $L_1$.)

We now have three things associated to $S_3$, namely $\zeta_{\mathbb{Q}}$, $L_1$ and $L_2$, and more complicated things are made from them. In particular, $\zeta_F$ has 1 copy of the "trivial" $\zeta_{\mathbb{Q}}$, 1 copy of $L_1$ and $2$ copies of $L_2$. That already screams representation theory to me. Then when I notice that the degree $d$ representations gives degree $d$ polynomials in $p^{-s}$ in the denominator, what could be more natural than a characteristic polynomial?