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!)

Advertisements