= Local parameter on a smooth algebraic curve
{title2=$t\in\mathfrak m\setminus\mathfrak m^2$}
= Local parameter
{synonym}
At a point of a <smooth algebraic curve>, its <local ring> is a <discrete valuation ring>. A local parameter is a generator $t$ of its <maximal ideal>, equivalently an element of <valuation> one. It measures the order of a <zero of a function> or <pole> by expressing functions as powers of $t$ times units. Over the complex numbers its analytic counterpart is a <local coordinate> vanishing simply at the point. For an <elliptic curve> in a <Weierstrass equation of an elliptic curve>, $t=-x/y$ is the standard parameter at the identity.
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