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.

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

#### Ramification

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

Let . Let be an element of the function field that generates the maximal ideal of the local ring (also a dvr) . Let be the -order of as a function of . Then is **ramified at** if and **unramified** if for every point .

We have the following :

#### Theorem:

Let be a non-constant map of curves. Then,

- For every point ,

- For all but finitely many points ,

where denotes the degree of separability of .

- If is another non-constant map of curves, then

#### Corresponding results in number theory

The first result corresponds to the identity for number fields . 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 be number fields with . Let and be the corresponding rings of integers. Then,

- For every prime in , we have

where the ‘s are primes in . Then, holds, where is the inertial degree given by .

- At most finitely many primes of ramify in . (A prime of is said to ramify in if for some ).

- If is a number field containing , then for every prime of ,

#### 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 is the group of units)

is the exact sequence of degree-zero divisors

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!)

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