Local parameter: Difference between revisions

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In the geometry of complex [[algebraic curvescurve]]s, a '''local parameter''' for a curve ''C'' at a (smooth) point ''P'' is just a rational[[meromorphic function]] on ''C'' that has a [[simple zero]] at ''P''. TheyThis areconcept usedcan mainlybe 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 ''countP'' of an algebraic curve properly''C'' (indefined over an [[algebraically closed field]]) is always a local[[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''.

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''.

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

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>\operatorname{ord}_P(f)=\max\{d=0,1,2,\ldots: f\in m^d_P\}</math>).

==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

<math>\operatorname{ord}_P(f)=\max\{d=0,1,2,\ldots: f\in m^d_P\};</math>

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 <math>\mathcal{O}_{C,P}</math>. Hence the concept of ''having a simple zero at a point P'' is now complete: it will be a rational function <math>f\in K(C)</math> such that its germ falls into <math>m_P^d</math>, with ''d'' at most 1.

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 a local parameter at ''P'' will be a uniformizing parameter for the DVR (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>), whence the name.

==Definition==

Let ''C'' be an algebraic curve (defined over an algebraically closed field ''K'')., Theand '''valuation''' onlet ''K''(''C'') (be the field of rational functions of ''C''. The '''valuation''' on ''K''(''C'') corresponding to a smooth point <math>P\in C</math> is defined as

<math>\operatorname{ord}_P(f/g)=\textoperatorname{ord}_P(f)-\operatorname{ord}_P(g)</math>, where <math>\operatorname{ord}_P</math> is the usual valuation on the local ring (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>). A '''local parameter''' for ''C'' at ''P'' is a function <math>t\in K(C)</math> such that <math>\operatorname{ord}_P(t)=1</math>.

==See also==