I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correct, but he doesn't give references, and the thought of ploughing through Artin's collected works seems a bit daunting to me, so I thought I'd ask here first.
Background.
If $V$ is a smooth (affine or projective) curve over a finite field $k$ of size $q$, then $k$ has (up to isomorphism) a unique extension $k_n$ of degree $n$ over $k$ (so $k_n$ has size $q^n$) and one can define $N_n$ to be the size of $V(k_n)$. Completely concretely, one can perhaps imagine the case where $V$ is defined by one equation in affine or projective 2-space, for example $y^2=x^3+1$ (note that this equation will give a smooth curve in affine 2-space for $p$, the characteristic of $k$, sufficiently large), and simply count the number of solutions to this equation with $x,y\in k_n\ $to get $N_n$. Let $F_V(u)=\sum_{n\geq1}N_nu^n$ denote the formal power series associated to this counting function.
Now it turns out from the "formalism of zeta functions" that this isn't the most ideal way to package the information of the $N_n$, one really wants to be doing a product over closed points of your variety. If $C_d$ is the number of closed points of $V$ of degree $d$, that is, the number of closed points $v$ of (the topological space underlying the scheme) $V$ such that $k(v)$ is isomorphic to $k_d$, then one really wants to define
$$Z_V(u)=\prod_{d\geq1}(1-u^d)^{-C_d}.$$
If one sets $u=q^{-s}$ then this is an analogue of the Riemann zeta function, which is a product over closed points of $Spec(\mathbf{Z})$ of an analogous thing.
Now the (easy to check) relation between the $C$s and the $N$s is that $N_n=\sum_{d|n}dC_d$, and this translates into a relation between $F_V$ and $Z_V$ of the form
$$uZ_V'(u)/Z_V(u)=F_V(u).$$
This relation also means one can compute $Z$ given $F$: one divides $F$ by $u$, integrates formally, and then exponentiates formally; this works because $f'/f=(\log(f))'$.
The reason I'm saying all of this is just to stress that this part of the theory is completely elementary.
The Weil conjectures in this setting.
The Weil conjectures imply that for $V$ as above, the power series $Z_V(u)$ is actually a rational function of $u$, and make various concrete statements about its explicit form (and in particular the location of zeros and poles). Note that they are usually stated for smooth projective varieties, but in the affine curve case one can take the smooth projective model for $V$ and then just throw away the finitely many extra points showing up to see that $Z_V(u)$ is a rational function in this case too.
How to prove special cases in 1923?
OK so here's the question. It's 1923, we are considering completely explicit affine or projective curves over explicit finite fields, and we want to check that this power series $Z_V(u)$ is a rational function. Dieudonne states that Artin manages to do this for curves of the form $y^2=P(x)$ for "many polynomials $P$ of low degree". How might we do this? For $P$ of degree 1 or 2, the curve is birational to projective 1-space and the story is easy. For $V$ equals projective 1-space, we have
$$F_V(u)=(1+q)u+(1+q^2)u^2+(1+q^3)u^3+\ldots=u/(1-u)+qu/(1-qu)$$
from which it follows easily from the above discussion that
$$Z_V(u)=1/[(1-u)(1-qu)].$$
For polynomials $P$ of degree 3 or 4, the curve has genus 1 and again I can envisage how Artin could have approached the problem. The curve will be birational to an elliptic curve, and it will lift to a characteristic zero curve with complex multiplication. The traces of Frobenius will be controlled by the corresponding Hecke character, a fact which surely will not have escaped Artin, and I can believe that he was now smart enough to put everything together.
For polynomials of degree 5 or more, given that it's 1923, the problem looks formidable.
Q1) When Dieudonne says that Artin verified (some piece of) the Weil conjectures for "many polynomials of low degree", does he mean "of degree at most 4", or did Artin really move into genus 2?
How much further can we get in 1931?
Now this one really surprised me. Dieudonne claims that in 1931 F. K. Schmidt proved rationality of $Z_V(u)$, plus the functional equation, plus the fact that $Z_V(u)=P(u)/(1-u)(1-qu)$, for $V$ an arbitrary smooth projective curve, and that he showed $P(u)$ was a polynomial of degree $2g$, with $g$ the genus of $V$. This is already a huge chunk of the Weil conjectures. We're missing the statement that $P(u)$ has all its rots of size $q^{-1/2}$ (the "Riemann hypothesis") but this is understandable: one needs a fair amount of machinery to prove this. What startled me (in my naivity) was that I had assumed that all this was due to Weil in the 1940s and I am obviously wrong: "all Weil did" was to prove RH. So I have a very basic history question:
Q2) However did Schmidt do this?
EDIT: brief summary of answers below (and what I learned from chasing up the references):
A1) Artin didn't do anything like what I suggested. He could explicitly compute the zeta function of an arbitrary given hyperelliptic curve over a given finite field by an elegant application of quadratic reciprocity. See e.g. the first of Roquette's three papers below. The method in theory works for all genera although the computations quickly get tiresome.
A2) Riemann-Roch. Express the product defining $Z$ as an infinite sum and then use your head.
Best Answer
Peter Roquette has written four beautiful papers on the history of the zeta-function in characteristic $p$.
The Riemann hypothesis in characteristic p, its origin and development. Part 1. The formation of the zeta functions of Artin and F.K. Schmidt.
The Riemann hypothesis in characteristic p, its origin and development. Part 2. The first steps by Davenport and Hasse.
The Riemann hypothesis in characteristic p, its origin and development. Part 3: The elliptic case.
The Riemann hypothesis in characteristic p, its origin and development. Part 4: Davenport-Hasse fields.
Relevant to your questions is part 1. From the abstract "This Part 1 is dealing with the development before Hasse's contributions to the Riemann hypothesis. We are trying to explain what he could build upon. The time interval covered will be roughly between 1921 and 1931. We start with Artin's thesis of 1921 where the Riemann hypothesis for function fields was spelled out and discussed for the first time, namely in the case of quadratic function fields. We will describe the activities following Artin's thesis until F.K.Schmidt's classical paper 1931 on the Riemann-Roch theorem and the zeta function of an arbitrary function field. Finally we will review Hasse's paper in 1934 where he gives a summary about all what was known at that time about zeta functions of function fields. "