Local parameter: Difference between revisions

Revision as of 02:46, 17 July 2009

Revision as of 13:14, 17 July 2009

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A local parameter is a generator of the unique maximal ideal associated to a local ring.

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]]s), due to the fact that 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''.