# L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the characteristic at $\mathfrak{p}$, there is an action of $\hat{\mathbb{Z}}$ on the inertia invariants $H^m_{et}(X_{\bar{k}},\mathbb{Q}_l)^I$ and we can define $$L_{\mathfrak{p}}(H^m(X),s)=\text{det}(1-f_{\mathfrak{p}}\cdot N\mathfrak{p}^{-s}|H^m_{et}(X_{\bar{k}},\mathbb{Q}_l)^I),$$ where $f_{\mathfrak{p}}\in\hat{\mathbb{Z}}$ is the geometric Frobenius. In some half plane, The Euler product over all $\mathfrak{p}$ is the mth Hasse-Weil zeta function: $$L(H^m(X),s)=\prod_{\mathfrak{p}}(H^m(X),s).$$ Their alternating product is the Hasse-Weil zeta function, which is closely related to the Hasse Zeta function of a proper regular model $\mathfrak{X}$ of $X$ (which is a scheme of finite type over $\mathbb{Z}$). The former only depends on $X$, the generic fiber of $\mathfrak{X}$, and the latter depends also on the special fibers, especially at the bad primes. For convenience the definitions are: $$\zeta_X(s)=\frac{\prod_{m=0}^{\text{dim(X)}}L(H^{2m}(X),s)}{\prod_{m=0}^{\text{dim}(X)-1}L(H^{2m-1}(X),s)}$$ $$\zeta_{\mathfrak{X}}(s)=\prod_{x\in\mathfrak{X}_0}\frac{1}{1-|k(x)|^{-s}},$$ where $\mathfrak{X}_0$ denotes the closed points of $\mathfrak{X}$ and $|k(x)|$ denotes the residue field at $x$. Again, both are defined in some half plane. Of course, we expect meromorphic continuation of both zeta functions, and by considering their quotient, which is a finite product over bad primes, it is easy to see that their continuations are equaivalent.

Moreover, we expect a functional equation of the Hasse zeta function of $\mathfrak{X}$, the arithmetic scheme, with respect to $s\mapsto\text{dim}(\mathfrak{X})-s$. This should be equivalent to the functional equations of each Hasse-Weil $L$ or, indeed, of the Hasse-Weil zeta.

I have two questions, the first of which is how does one modify the conductor of $X$ to deduce an appropriate conductor for $\mathfrak{X}$? I know the process for curves and arithmetic surfaces.

The second question is, is it possible to prove that the quotient $\zeta_{\mathfrak{X}}(s)/\zeta_X(s)$ is invariant with respect to $s\mapsto\text{dim}(\mathfrak{X})-s$? Again, I understand the argument for curves.

• I'm excited to see an answer to this. I've tried to look this type of thing up several times with no success. I imagine this issue about the conductor is still completely open in higher dimensions because something really subtle is going on that has to do with the fact that your model could have bad reduction somewhere but still an unramified $\ell$-adic representation. Even if $E$ is an elliptic curve and the class number of $k$ is not $1$ I think you'll have trouble "detecting the right conductor" from your global model... – Matt Oct 8 '13 at 17:27
• For curves the modification needed is explained in Bloch "DeRham cohomology and conductors of curves". Indeed, it involves some consideration of the quotient from my second question. I have no idea where to start for dimension 2 or more, but I figured it made sense to start there. – Tom163 Oct 9 '13 at 20:08