In this post, we shall see some results about algebraic curves. For the past week, I have been studying algebraic curves and the great Riemann-Roch theorem. I won’t go in the details but will only sketch the basic theory behind their study.

Algebraic curves

By a curve we will mean a projective variety of dimension 1. For the sake of simplicity let us assume the base field {k} to be algebraically closed. (This is far from sufficient; indeed the most interesting applications in number theory will have {k} as the rational numbers or finite fields or {p}-adic fields). One can think of the curve as the locus of the zero set of an irreducible polynomial {F(X,Y) \in k[X,Y]}. This as it is, is an affine curve and we attach points of infinity as necessary by homogenizing {F}. Note that it is not necessary that the curve be generated by just one polynomial. It may be possible that it is the intersection of higher dimensional varieties in projective space of higher dimension. The only requirement is that the dimension of this variety be 1.

Function field of a curve

Given a curve {C}, we can associate a field to it, namely the function field of {C}. As a concrete example, let {C} be the projective circle given by the homogeneous equation {F(X,Y,Z) = X^2 + Y^2 - Z^2}. (If you are not habituated to using projective coordinates, just substitute 1 for {Z} and everything should work fine). Then the function field of {C} is

\displaystyle k(C) = \text{Field of fractions of } \displaystyle\frac{k[X,Y,Z]}{(X^2+Y^2-Z^2)}.

Note that asking the dimension of {C} to be 1 is the same as {k9C)} having a transcendence degree 1 over {k}.

Corresponding to any point {P=[a:b:c]} on the curve, there is a discrete valuation ring {k[C]_P} which is the localization of the domain {\displaystyle\frac{k[X,Y,Z]}{(X^2+Y^2-Z^2)}} at the maximal ideal {(X-a, Y-b, Z-c)}. A fundamental theorem in algebraic geometry says that the point {P} on {C} (in fact any variety) is non-singular if and only if {k[C]_P} is a regular local ring.

Maps between curves

By a map between curves {C_1} and {C_2} we will mean a morphism of the corresponding projective varieties. If

\displaystyle \phi : C_1 \rightarrow C_2

is a morphism of curves, then {\phi} is either constant or surjective! Also, {\phi} induces a map between the function fields viz.

\displaystyle \phi^* : k(C_2) \hookrightarrow k(C_1) \qquad \phi^*(f) = f \circ \phi.

If {\phi} is nonconstant this gives a finite extension of fields, {[k(C_1) : \phi^*(k(C_2))]} and we define the degree {deg(\phi)} to be the degree of this extension. A map of degree 1 is an isomorphism.

Categorical equivalence between curves and function fields

We saw that a curve {C} gives a function field {k(C)} of transcendence degree 1 over {k}. Morphisms of curves give an inclusion of function fields. Indeed, this functoriality goes beyond, it’s a categorical equivalence. Given a field {\mathbb K/k} of transcendence degree 1, one proves that the collection of local rings {R} such that {k \subset R \subset \mathbb K} actually define a non-singular projective curve. (All the {R}‘s will be dvrs since tr. deg{(\mathbb K/k)=1}). The morphisms in this category are inclusion maps of fields and they give morphisms of curves.

If you have read this far, then you are very close to understanding the connection between algebraic curves and algebraic number fields. This is explained in the next post here.

References

  • R. Hartshorne, Algebraic Geometry (Chapter 1)
  • J. Silverman, Arithmetic of Elliptic Curves (Chapter 2)
Advertisements