Local parameter: Difference between revisions

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In the geometry of [[algebraic curves]], a '''local parameter''' for a curve ''C'' at a (smooth) point ''P'' is a rational function that has a simple zero at ''P''. They are used mainly to ''count properly'' (in a local way).

This has an algebraic resemblance with the concept of a [[Discrete_valuation_ring#Uniformizing_parameter|uniformizing parameter]] (or just '''uniformizer''') found in the context of [[discrete valuation ring]]s 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 the local ring at a smooth point of an algebraic curve is always a discrete valuation ring.<ref>J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. p. 21</ref> If this is the case of <math>P\in C</math>, then the maximal ideal of <math>\mathcal{O}_{C,P}</math> consists of all those regular functions defined around ''P'' which vanishes at ''P'', and finally a local parameter at ''P'' will be a uniformizing parameter for the DVR (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>) (where the valuation function is given by <math>\text{ord}_P(f)=\text{max}\{d=0,1,2,...\dots: f\in m^d_P\}</math>).

==Definition==