# Higher-dimensional Artin L-functions

I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.

Now for the set-up. This will be very similar to the set-up of Chapter 9 of Serre's book "Lectures on N_X(p)". Let $R$ be a finite type flat $\mathbb{Z}$-algebra of dimension $n$ with fraction field $K$ and let $X = \mathrm{Spec}(R)$. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $Y$ be the normalisation of $X$ in $L$. At any point I am quite happy to shrink $X$. For example, I may assume that $X$ is regular and that $Y/X$ is finite étale.

Now for the analogue of Artin L-functions. For a complex representation $\rho$ of $G$ we define $$L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1}, \quad \text{re}(s) > n.$$ See the above book of Serre for the definition of Frobenius elements in this setting. That the above Euler product converges in the range $\text{re}(s) > n$ follows from the Weil conjectures.

My first question is the following.

1. Have such L-functions been studied before?

It seems that Deligne's Weil II paper should be relevant here, but alas I don't understand it well enough.

But what I would really like to know is the following.

1. Assume that $\rho$ is irreducible and not the trivial representation. Then does $L(s,\rho)$ admit a holomorphic continuation to the line $\text{re}(s) = n$ without zeros?

This latter fact is proved in the classical case where $n = 1$ by using Brauer's induction theorem to reduce to the case where $G$ is abelian. Here one essentially obtains a dirichlet L-function, where the result is well-known (being part of the proof of Dirichlet's theorem on primes in arithmetic progressions). I imagine a similar reduction to the abelian case can be made in my setting, but of course the point is that I don't know how to deal with the abelian case itself.

• There is an article of Serre on zeta functions associated to schemes $X$ of finite type over $\mathbb{Z}$. If I recall correctly, if $G$ is a finite group acting on $X$ by automorphisms, and $\chi$ is a complex character of $G$, then you can define a zeta function $\zeta(X,\chi,s)$. Does this give the same $L$-function you suggest to look at? – François Brunault Oct 15 '15 at 9:13
• Quite possibly. Which article are you referring to? Serre has many... – Daniel Loughran Oct 15 '15 at 9:17
• I don't have Serre's collected papers at hand, but I think the article in question is MR0194396 (33 #2606) Serre, Jean-Pierre . Zeta and L functions. 1965 Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) pp. 82--92 Harper & Row, New York. – François Brunault Oct 15 '15 at 9:20
• Thanks for the reference. I have only managed to find the paper so far in Russian.... I will keep on looking. – Daniel Loughran Oct 15 '15 at 14:12

According to Serre in "Zeta and L-functions" (MR0194396), these L-functions have been defined by Artin himself. Moreover, it seems that in the formula $$L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1}$$ one should define $\rho(\mathrm{Frob}_x)$ as the average of the $\rho(g)$ for $g \in G$ mapping to $\mathrm{Frob}_x$.
Serre further says that $L(s,\rho)$ can be continued to a meromorphic function in the half-plane $\mathrm{Re}(s)>n-\frac12$, and that the singularities « can be determined, or rather reduced to the classical case $n=1$ » (there is an interesting lemma in the case $Y \to Y'$ is a fibration of curves, enabling to do induction on the dimension of $Y$). I don't know whether this answers your Question 2.
• Thanks very much for the reference (I have now found a copy of it in English). In fact Serre mentions a result in his paper which is almost exactly the answer to Q2. Namely the final Corollary in the paper states that $L(s,\rho)$ is holomorphic and non-zero at $s=n$. I wonder if it is somehow implicit in this work that $L(s,\rho)$ is also holomorphic and non-zero on the line $\text{re}(s) = n$? – Daniel Loughran Oct 15 '15 at 16:20
• It seems that Serre's inductive argument on the dimension to show $L(\rho,n) \neq 0$ could also be used on the line $\mathrm{Re}(s)=n$, but I'm really no expert here. Unfortunately I know of no other reference than this article. It would be indeed very nice to have a reference with more detailed proofs. – François Brunault Oct 15 '15 at 17:34
• It would be also very nice to define a $G$-equivariant L-function in this setting. I have not seen a reference for this. This would not really be a new object, but this would tie things together nicely I think. – François Brunault Oct 15 '15 at 17:39