# Local parameter

In the geometry of complex algebraic curves, a **local parameter** for a curve *C* at a smooth point *P* is just a meromorphic function on *C* that has a simple zero at *P*. This concept can be generalized to curves defined over fields other than
(or even schemes), because the local ring at a smooth point *P* of an algebraic curve *C* (defined over an algebraically closed field) is always a discrete valuation ring.[1] This valuation will endow us with a way to count the order (at the point *P*) of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) having a zero or a pole at *P*.

Local parameters, as its name indicates, are used mainly to properly *count multiplicities* in a local way.

## Introduction

When *C* is a complex algebraic curve, we know how to count multiplicities of zeroes and poles of meromorphic functions defined on it.[2] However, when discussing curves defined over fields other than
, we do not have access to the power of the complex analysis, and a replacement must be found in order to define multiplicities of zeroes and poles of rational functions defined on such curves. In this last case, we say that the germ of the regular function
vanishes at
if
. This is in complete analogy with the complex case, in which the maximal ideal of the local ring at a point *P* is actually conformed by the germs of holomorphic functions vanishing at *P*.

Now, the valuation function on is given by

this valuation can naturally be extended to *K*(*C*) (which is the field of rational functions of C) because it is the field of fractions of
. Hence the idea of *having a simple zero at a point P* is now complete: it will be a rational function
such that its germ falls into
, with *d* at most 1.

This has an algebraic resemblance with the concept of a uniformizing parameter (or just **uniformizer**) found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR (*R, m*) is just a generator of the maximal ideal *m*. The link comes from the fact that a local parameter at *P* will be a uniformizing parameter for the DVR (
,
), whence the name.

## Definition

Let *C* be an algebraic curve defined over an algebraically closed field *K*, and let *K*(*C*) be the field of rational functions of *C*. The **valuation** on *K*(*C*) corresponding to a smooth point
is defined as
, where
is the usual valuation on the local ring (
,
). A **local parameter** for *C* at *P* is a function
such that
.

## See also

## References

- J. H. Silverman (1986).
*The arithmetic of elliptic curves*. Springer. p. 21 - R. Miranda (1995).
*Algebraic curves and Riemann surfaces*. American Mathematical Society. p. 26