I have to give a presentation about function fields and valuation.
I will discuss the case of rational function on the projective line. Then I would like to give an example of a different function field. I would like to talk about the function field on an elliptic curve to show the difference with rational standard function, but I do not know how to briefly explain them the concept of valuation at a point.
Let's consider $y^2=x^3+1$. How can I define a valuation at a point of the curve without using much algebra like local rings and valuations? I would like to use the definition used in the rational case: a polynomial $p(x) $ in the regular function ring has a zero of order n at the point $(a,b)$ if $p(x)=(x-a)^nh(x,y)$, but the fact that I am working in a quotient ring gives some ambiguity in the choice of $n$ and $h$.
Do you now some different ways to present this topic?
For an elliptic curve over (a subfield of) $\Bbb{C}$ the easiest way is to expand the square root in binomial series.
Let $$E = \{ (x,y)\in \Bbb{C}^2, y^2=x^3+1\}$$ around $P=(0,1)\in E$ we get $$y =(1+ x^3)^{1/2}=\sum_{k=0}^\infty {1/2\choose k}x^{3k}$$ which is a convergent series for $|x|$ small enough.
Thus the function field is the fraction field of the ring of convergent power series $$\Bbb{C}[x,\sum_{k=0}^\infty {1/2\choose k}x^{3k}]$$ whose non-zero elements are of the form $f(x)=x^n \frac{\sum_{m\ge 0} c_m x^m}{\sum_{n\ge 0} d_n x^n},c_0\ne 0,d_0\ne 0$ to which we assign the non-archimedian absolute value $|f|_P = 2^{-n}$ associated to the discrete valuation $v_P(f)=n$.
Note $\frac{1}{\sum_{n\ge 0} d_n x^n}$ can be expanded in a power series near $x=0$ which gives an embedding of the function field into the ring of Laurent series $\Bbb{C}[[x]][x^{-1}]$.
The same process holds over any field once we replace the convergent series by formal power series : discrete valuation on a curve $=$ embedding of the function field into $k[[t]][t^{-1}]$.