- 0-geometry: Curves
- 0-geometry: Genus
- 0-geometry: Hurwitz
- ABC-theorem for Curves
In this series we collect our rough prep-notes for the lectures ahead. We focus on the main ideas and give precise references. More complete course-notes may follow afterwards.
Fix a perfect field $k$ (say a finite field) with algebraic closure $\overline{k}$ and absolute Galois group $G = \operatorname{Gal}(\overline{k}/k)$.
Aim: to study smooth projective $k$-curves via their function fields. This will allow us later to associate ‘curves’ to number fields.
We need two categories:
$\mathsf{Curves}/k$
The objects are smooth projective algebraic curves defined over $k$ (that is, a smooth closed subvariety $C$ of dimension one of some projective space $\mathbb{P}^n(\overline{k})$ defined by a set of homogeneous polynomials all of their coefficients belonging to $k$). We will call such objects curves defined over $k$.
The morphisms will be surjective algebraic maps $C \to C’$ defined over $k$ (that is, all coordinate functions have their coefficients in $k$). Remember that any non-constant rational map between two curves is automatically surjective. We will call such morphisms covers.
$\mathsf{1Fields}/k$
The objects are field extensions $K$ of $k$ of transcendence degree one with $k$ as their ‘field of constants’. That is, $K \cap \overline{k} = k$.
The morphisms will be field inclusions $K \hookrightarrow K’$ fixing $k$.
Main result: These categories are (anti)-equivalent to each other.
Details are in section I.6 of Robin Hartshorne’s Algebraic Geometry when $k = \overline{k}$ and modifications for the general case are in section II.2 of Joseph Silverman’s The Arithmetic of Elliptic Curves.
Sketch of proof: The direction from curves to fields is straightforward.
The contravariant functor $\mathsf{Curves}/k \longrightarrow \mathsf{1Fields}/k$ assigns to a curve $C$ its function field $k(C)$ (the field consisting of all rational functions $f:~C \rightarrow \overline{k}$ defined over $k$).
This functor associates to a cover $\phi:~C \mapsto C’$ the field-inclusion $\phi^{\ast}:~k(C’) \rightarrow k(C)$ obtained by composition (that is, $\phi^{\ast}(f) = f \circ \phi~:~C \rightarrow \overline{k}$ for all $f \in k(C’)$).
Conversely, the contravariant functor $\mathsf{1Fields}/k \longrightarrow \mathsf{Curves}/k$ assigns to a field $K$ of transcendence degree one
- the geometric points $C(\overline{k})$ of the curve $C$, which is the set of all discrete valuations rings in $K \otimes \overline{k}$ with residue field $\overline{k}$. The Galois group $G$ acts on this set, and,
- the schematic points of $C$ are the $G$-orbits of this action. Equivalently, these are the discrete valuation rings of $K$ with residue field a finite field extension $L$ of $k$. The degree of such a scheme-point is the size of the $G$-orbit (or the $k$-dimension of the residue field $L$ of the discrete valuation ring).
Example: Under this equivalence, the purely transcendental field $k(x)$ corresponds to the projective line $\mathbb{P}^1$ over $k$. Its geometric points $\mathbb{P}^1(\overline{k})$ are the points
$$\{ [\alpha : 1]~:~\alpha \in \overline{k} \} \cup \{ \infty = [1 : 0] \}$$
The discrete valuation ring of $\overline{k}(x)$ corresponding to $[\alpha : 1]$ has uniformizing parameter $x-\alpha$ and the one corresponding to $\infty$ has uniformising parameter $\tfrac{1}{x}$.
The Galois group fixes $\infty$ and acts on the point $[\alpha : 1]$ as it does on $\alpha \in \overline{k}$. Hence, the schematic points of $\mathbb{P}^1$ are $\infty$ together with all irreducible monic polynomials in $k[x]$.
Under the equivalence, the set of all non-constant maps $C \mapsto \mathbb{P}^1$ corresponds to the set of all $k$-field morphisms $k(x) \hookrightarrow k(C)$ and as these are determined by the image of $x$ they are determined by $f \in k(C)$. The cover corresponding to $f$
$$C(\overline{k}) \mapsto \mathbb{P}^1(\overline{k}) \quad \text{maps} \quad P \mapsto [f(P) : 1]$$
if $f$ is regular in $P$ and to $\infty$ otherwise.
