# The First Reciprocity Law

This is a follow up to this post where I defined the Abel-Jacobi map of a compact Riemann surface of genus to its jacobian . My first goal will be to demonstrate Abel’s theorem, which goes like this:

Theorem (Abel):For and a basis for the space of holomorphic 1-forms, the divisor is principal, i.e. for some meromorphic function if and only if , i.e. iff.

As a corollary, we will have that is in fact a smooth analytic embedding of into its Jacobian. Before proving the theorem, we will establish in this post a **reciprocity law** relating the periods of a holomorphic 1-form and a meromorphic 1-form having only simple poles. These two types of 1-forms are classically called **differentials of the first and third kind**, respectively. Recall the notion of **canonical basis** for from last post.

Proposition (First Reciprocity Law):Let be cycles inducing a canonical basis of and suppose are respectively differential of the first and third kind. Let be the poles of . Then the following relation holds:.

On the right hand side, the integral is taken on any path inside which is simply connected and is any base point.

**Proof:** The region is the standard polygonal representation of the Riemann surface , which is a -sided plane polygonal with sides labelled etc… with the corresponding orientation. This is hard to convey without drawing pictures so you should refer to some book if you’ve never seen this, maybe a book which treats the classification of closed surfaces like Munkres. Think of the standard way to represent a torus as a quotient of the square (which is a -polygonal when ) but with more sides.

Anyways, since is simply connected, we can choose and define without ambiguity the function

for . Then is a holomorphic mapping on the closure of with . By construction, if and are two points of that are identified in on the cycle , then

and similarly for and that are identified in , we have

.

Using this, we find that

and similarly

.

Moreover, by the residue theorem, we have

.

Equating these two last equations, we obtain the first reciprocity law.

QED

A first consequence of this result is that it permits us to choose a very nice basis for the holomorphic 1-forms . This will be a consequence of the

Corollary 1:For a non-zero holomorphic 1-form, we have.

**Proof:** Take . Then

for like above. So

.

So by integrating like above, we find

so letting gives the result since is positive.

QED

A **normalised**** basis** for with respect to the basis for will be a basis such that is if and if not. Corollary 1 permits us to always choose a normalised basis. Indeed, consider the linear mapping defined by

.

Then iff for all which by corollary 1 would mean .

Recall from last post the **period matrix** defined by where the columns are the vectors translated. By choosing a normalised basis, the period matrix becomes

for some matrix consisting of the **B-periods**.

Corollary 2 (Riemann’s bilinear relations):

1) First bilinear relation:If are two differentials of the first kind (i.e. holomorphic), then the reciprocity law tells us.

In particular, with a normalised basis,

,

i.e. is a symmetric matrix.

2) Second bilinear relation:Im, i.e. the matrix consisting of the imaginary parts of the coefficients of is positive definite.

**Proof:** The first bilinear relation is immediate from the first reciprocity law. For the second, we proceed as in the proof of the first corollary and write

.

But by the first bilinear relation this last expression is equal to

.

QED

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