Local parameter: Difference between revisions

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{{Orphan|date=December 2012}}

In the geometry of complex [[algebraic curve]]s, 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 <math>\mathbb{C}</math> (or even [[scheme (mathematics)|scheme]]s), 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]].<ref>J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. p. 21</ref> 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''.

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==Introduction==

When ''C'' is a complex algebraic curve, we know how to count multiplicities of zeroes and poles of meromorphic functions defined on it.<ref>R. Miranda (1995). ''Algebraic curves and Riemann surfaces''. American Mathematical Society. p. 26</ref>. However, when discussing curves defined over fields other than <math>\mathbb{C}</math>, 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 <math>f</math> vanishes at <math>P\in C</math> if <math>f\in m_P\subset\mathcal{O}_{C,P}</math>. 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 <math>\mathcal{O}_{C,P}</math> is given by