Is there a unified description (or a set of axioms) of the zeta function of an algebraic curve over a finite field $\mathbb{F}_q$ and the Riemann zeta function?
$\begingroup$
$\endgroup$
3
-
2$\begingroup$ Have a look at J-P Serre, "Facteurs locaux des fonctions zêta...", Séminaire Delange-Pisot-Poitou, tome 11, no. 2 (1969-1970), exp. 19. (available on NUMDAM). In the setting of Serre, the Riemann zeta function is essentially the zeta function of the projective line over $\bf Q$. $\endgroup$– Damian RösslerCommented Jul 30, 2012 at 22:41
-
1$\begingroup$ @Damian: of the projective line over $\mathbf{Q}$ ? I had always thought that it was the zeta function of $\mathrm{Spec}(\mathbf{Z})$. $\endgroup$– Chandan Singh DalawatCommented Jul 31, 2012 at 3:31
-
$\begingroup$ @Chandan Singh Dalawat. In fact it is $\zeta(s)\zeta(s-1)$, where $\zeta(s)$ is the Riemann zeta function. The point I wanted to make is that zeta functions are defined for varieties over a global field, which is either a number field or the function field of an algebraic curve over a finite field. A simpler example is of course $\bf Q$ itself, which gives the Riemann zeta function, as you say. $\endgroup$– Damian RösslerCommented Jul 31, 2012 at 5:57
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
1
Let $D$ be a ring which is infinite, not a field, and in which $D/I$ is finite for any nonzero ideal $I$ (what Pete Clark calls a abstract number ring). Then we can define $N(I) = |D/I|$ and write down a zeta function $$\zeta_D(s) = \sum_{I \neq 0} \frac{1}{N(I)^s}.$$
This specializes to the Riemann zeta function when $D = \mathbb{Z}$, to the Dedekind zeta function when $D = \mathcal{O}_K$, and to the zeta function of an affine algebraic curve over a finite field when $D$ is its ring of functions. If $D$ is Dedekind, then unique factorization of ideals gives an Euler product for this zeta function.
To get the zeta function of a curve, not necessarily affine, we replace ideals with effective divisors, that is, non-negative formal integer linear combinations $$\sum n_i P_i, n_i \ge 0$$
of points of $C$ over $\overline{\mathbb{F}_q}$ which are $\text{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$-invariant, and the appropriate replacement for the norm here is $q^{\sum n_i}$. This reduces to the above description in the case of a (nonsingular, geometrically integral) affine curve. (The more standard expression for the zeta function of a curve over a finite field, which tells you what the logarithm of the zeta function is in terms of counting points over finite extensions of $\mathbb{F}_q$, can be shown to be equivalent to this description using the exponential formula in combinatorics. See this blog post.)
answered Jul 30, 2012 at 22:48
-
3$\begingroup$ Fun fact: today I am thinking about counting ideals in the coordinate ring of an affine curve over a finite field. Searching the internet for the fact I needed about the zeta function, I came to your answer, which I found helpful. Then I clicked on the link to my own set of notes, which was further help! Thanks for the answer. $\endgroup$ Commented May 16, 2013 at 15:53
$\begingroup$
$\endgroup$
If $X$ is a $\mathbb{Z}$-scheme of finite type, its zeta function is the product $\zeta_X(s)= \prod_{x \in |X|}(1-|k(x)|^{-s})^{-1}$ where $|X|$ denotes the set of closed points of $X$ and $|k(x)|$ is the cardinality of the (finite) residue field of $\mathcal{O}_{X,x}$. When $X=\mathrm{Spec}\mathbb{Z}$, this is the Riemann zeta function. When $X$ is a curve over a finite field $\mathbb{F}_q$, $\zeta_X(s)=Z(X,q^{-s})$ where $Z(X,t)=\exp{ \left(\sum |X(\mathbb{F}_{q^n})|\frac{t^n}{n} \right)}$ is the usual zeta function.
answered Jul 30, 2012 at 23:15