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In this post, I will show how algebraic curves and algebraic numbers are related. I shall continue using the notations developed in the previous post here. You may read that post to refresh the theory of algebraic curves.

The unifying feature between the two is the concerned rings are Dedekind domains. A Dedekind domain is an integrally closed Noetherian domain of dimension 1. It has two excellent properties,

  • The localization at every prime ideal is a dvr.
  • The integral closure of a DD in a finite separable extension of its field of fractions is again a DD.

In the curves case, suppose we have two curves (i.e., non-singular projective varieties of dimension 1) and a non-constant morphism between them.

\displaystyle \phi : C_1 \rightarrow C_2.

Then as seen earlier, it induces an inclusion of function fields,

\displaystyle \phi^* : k(C_2) \hookrightarrow k(C_1).

Ramification

We shall now define what ramification at a point {P \in C_1} means. Intuitively, it means there is a knot at {P}. Let us use some commutative algebra to make this notion precise.

Let {Q = \phi(P)}. Let {t} be an element of the function field {k(C_2)} that generates the maximal ideal of the local ring (also a dvr) {k[C_2]_Q}. Let {e_P} be the {P}-order of {\phi^*(t) = t \circ \phi} as a function of {k(C_1)}. Then {\phi} is ramified at {P} if {e_P>1} and unramified if {e_P=1} for every point {P\in C_1}.

We have the following :

Theorem:

Let {\phi : C_1 \rightarrow C_2} be a non-constant map of curves. Then,

  • For every point {Q\in C_2},

\displaystyle \text{deg } \phi = \displaystyle\sum_{P\mapsto Q} e_P.

  • For all but finitely many points {Q\in C_2},

\displaystyle |\phi^{-1}(Q)| = \text{deg}_s \phi,

where {\text{deg}_s} denotes the degree of separability of {k(C_1) / \phi^* (k(C_2))}.

  • If {\psi : C_2 \rightarrow C_3} is another non-constant map of curves, then

\displaystyle e_{\psi\circ\phi}(P) = e_\phi (P) . e_\psi (\phi P).

Corresponding results in number theory

The first result corresponds to the identity {\displaystyle\sum_{i=1}^g e_i f_i = [L:K]} for number fields {L/K}. The second one says that only finitely many primes ramify and the third result is the multiplicativity of ramification indices in a tower of number fields. Let us state these results more precisely in the following

Theorem:

Let {K \subseteq L} be number fields with {[L:K]< \infty}. Let {\mathcal O_K} and {\mathcal O_L} be the corresponding rings of integers. Then,

  • For every prime {P} in {\mathcal O_K}, we have

\displaystyle P \mathcal O_L = Q_1^{e_1} Q_2^{e_2} \cdots Q_g^{e_g}

where the {Q_i}‘s are primes in {\mathcal O_L}. Then, \displaystyle \displaystyle\sum_{i=1}^g e_i f_i = [L:K] holds, where {f_i} is the inertial degree given by {[ \mathcal O_L/Q_i : \mathcal O_K/P ]}.

  • At most finitely many primes of {\mathcal O_K} ramify in {\mathcal O_L}. (A prime {P} of {\mathcal O_K} is said to ramify in {L} if {e_i>1} for some {i}).
  • If {M} is a number field containing {L}, then for every prime {P} of {\mathcal O_K},

\displaystyle e_{M/K} = e_{M/L}. e_{L/K}.

More analogy

The similarity between number fields and algebraic curves does not end here. In the number theoretic case, we have the class group of a number field which is the quotient of the free abelian group on prime ideals modulo the principle ideals. Similarly, for algebraic curves we have the Picard group which is the free abelian group on divisors modulo principal divisors. Both groups turn to be finite (after some struggle in proving it).

Finally, the analog of the exact sequence in number theory (here {U_K} is the group of units)

\displaystyle 1 \rightarrow U_K \rightarrow K^* \rightarrow \displaystyle \begin{pmatrix} \text{fractional} \\ \text{ideals of }\mathcal O_K \end{pmatrix} \rightarrow C_K \rightarrow 0

is the exact sequence of degree-zero divisors

\displaystyle 1 \rightarrow K^* \rightarrow K(C)^* \rightarrow \text{Div}^0(C) \rightarrow \text{Pic}^0(C) \rightarrow 0.

They were the brilliant schemes of Grothendieck and his co-workers that unified algebraic geometry and number theory with tools (results) from the former being made available to the latter. He was able to prove the Weil conjectures with these abstract unified objects known as schemes but more on that later (after I study it!)

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)

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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