What does the power of an ideal *mean*? I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at $P$), namely
$$\operatorname{ord}_P\colon\;\bar K[C]_P\to\{0,1,2,\dots\}\cup\{\infty\}
\\\operatorname{ord}_P(f)=\sup\,\{\,d\in\mathbb Z\,:\,f\in M_P^d\,\}\text.$$
Here $M_P$ is an ideal of $\bar K[C]$, defined as
$$M_P=\{\,f\in\bar K[C]\,:\,f(P)=0\,\}\text.$$
Of course I am well aware how the involved operations are defined, I just don't have any "feeling" for them. What does it mean for a polynomial to be in $M_P^d$?
 A: Actually, valuation is a very geometric concept in the case of varieties and curves. The valuation of a function $f$ at a point (divisor, in the case of varieties) $P$ is the multiplicity of $P$ as a root of $f$: it is $>0$ if $P$ is a zero of $f$ and $<0$ if $P$ is a pole of $f$.
Another useful definition is to use a local parametrization $g(t)$ (by formal series) around $P$ (called a “uniformizer”, but think of implicit functions if you want), such that $g(0) = P$, $g'(0) \neq 0$, and $g(t)$ locally parametrizes the curve $C$. Then for any function $f$ on $C$, you may write the composed series $f(g(t))$ in the form
$$ f(g(t)) = \sum_{i \geq n} a_i t^i, \quad a_n \neq 0,$$
and the index $n$ of the first non-zero coefficient is exactly the valuation of $f$ at $P$.
Example: for the elliptic curve $y^2 = x^3 + x$ over the rationals and the point $P = (0,0)$, we may use the local parametrization $t \mapsto (x(t), t^2)$ where $x(t)$ is the inverse series to $x^3 + x$, given by $x(t) = t - t^3 + 3 t^5 - 12 t^7 + \dots$. (Check that this does give a point of the curve!).
Since $x(t) = t + \dots$, the valuation of $x$ at this point is $1$; since $y(t) = t^2$, the valuation of $y$ at this point is $2$. The important fact is that we could have chosen any other uniformizer (such as $x$ itself), the valuations would have remained the same.
