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 Feynmann'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?

  • 1
    $\begingroup$ You want the denominator in the Euler factor at $\mathfrak p$ to be a polynomial in $1/{\mathrm N}(\mathfrak p)^s$ of degree at most the degree of the representation (and usually "at most" should be "equal to"), and you want this polynomial associated to a matrix $\rho({\rm Frob}_{\mathfrak p})$ to be invariant under conjugation of the matrix. If you want a mapping from $d \times d$ matrices to polynomials of degree $d$ that is the same on conjugate matrices, and you have any experience with linear algebra, the obvious idea is to use the characteristic polynomial of the matrix. What else? $\endgroup$ – KCd Jun 3 '16 at 16:30
  • $\begingroup$ An ambiguity is whether to the matrix $A$ you want to associate the usual characteristic polynomial $\det(A - {\lambda}I_d)$ or its variants like $\det(\lambda{I}_d - A)$ or $\det(I_d - \lambda{A})$. Since we want to feed the results into an infinite product and have it converge, the best choice is to use polynomials with constant term 1, so we go with $\det(I_d - \lambda{A})$. $\endgroup$ – KCd Jun 3 '16 at 16:33
  • $\begingroup$ @KCd I would mention $$\log \det(I-\rho(\text{Frob}_\mathfrak{p}) N(\mathfrak{p})^{-s}) = \sum_{k=1}^\infty \frac{N(\mathfrak{p})^{-sk}}{k} \text{tr}(\rho(\text{Frob}_\mathfrak{p}^k))$$ (now for the ramifed primes it is a little more messy and I heard Artin spent a few years to solve it) $\endgroup$ – reuns Oct 6 '17 at 18:04

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