Again a question about algebraic varieties ! Actually, I followed to book of Silverman "The Arithmetic of elliptic curve", and I have several questions about ramification index. For $\phi : C_1 \to C_2$ a non constant map of smooth curves, and $P \in C_1$, he's defining the ramification index of $\phi$ at $P$ as : $e_{\phi}(P) = \operatorname{ord}_{P}(\phi^*(t_{\phi(P)}))$ where $t_{\phi(P)}$ is a uniformizer at $\phi(P)$. Then, we have a proposition among which we have the formula : $\forall Q \in C_2 \; \operatorname{deg}(\phi) = \sum_{P \in \phi^{-1}(Q)} e_{\phi}(P)$.
My problem is : how to explicitly calculate $\operatorname{deg}(\phi)$ using this formula. I mean, the author gives then an example :
$\phi : \mathbb{P}^1 \to \mathbb{P}^1 \; [X:Y] \mapsto [X^3(X-Y)^2:Y^5]$, and he says that $\phi$ is unramified everywhere except on $[0:1]$ and $[1:1]$ where we find : $e_{\phi}([0:1]) = 3, e_{\phi}([1:1]) = 2$. So, I tried to understand this example, but I'm stuck.
Actually, I didn't first saw why it is unramified everywhere except on $[0:1]$ and $[1:1]$, so I tried to understand what happens in $[0:1]$ and $[1:1]$ for example.
- If $Q=[a:b], \; b \neq 0$, we have : $\mathcal{O}_{\mathbb{P}^1, Q}= k[\frac{X}{Y}]_{(\frac{X}{Y}-\frac{a}{b})}$ with maximal ideal : $\mathcal{m}_q = (\frac{X}{Y}-\frac{a}{b})k[\frac{X}{Y}]_{(\frac{X}{Y}-\frac{a}{b})}$. So, for $Q=[0:1]$ for instance, we have : $\mathcal{O}_{\mathbb{P}^1, Q}= k[\frac{X}{Y}]_{(\frac{X}{Y})}$ with maximal ideal : $(\frac{X}{Y})k[\frac{X}{Y}]_{(\frac{X}{Y})}$, and a uniformizer is then given by $\frac{X}{Y}$. But we have : $$ e_{\phi}([0:1]) = \operatorname{ord}_{[0:1]}(\frac{X}{Y} \circ [X^3(X-Y)^2 : Y^5]) = \operatorname{ord}_{[0:1]}(\frac{X^3}{Y^3}(X-Y)^2.\frac{1}{Y^2}) $$ which is not in the maximal ideal, but : $$ (\frac{X^3}{Y^5}(X-Y)^2)^2 = \frac{X^6}{Y^6}.\frac{(X-Y)^4}{Y^4} $$ seems to be on the ideal cause it's a quotient of polynomial of same degree of the form : $\frac{X}{Y} \times (*)$. So, we find : $e_{\phi}([0:1]) = 2$ ? So, obviously I'm wrong somewhere, or there is something I didn't understood.
And it's the same for the other one. So my first question is : where I'm wrong, and how to explictly determine the ramification index ?
My second question is : if we know that $\phi : C_1 \to C_2$ is given by $[x:y] \mapsto [f_1(x) : 1]$ for example, and we know in some point $P$ $f_1$ as a pole of order $n$ fixed, and otherwise $f_1$ as neither a pole nor a zero. Can we conclude that : $deg(\phi)= – \operatorname{ord}_P(f_1)) = n$ and the same if we replace the pole by a zero ? Put an other way : is there a link between the pole and zeros of the rational functions defining the map and the degree of the map ?
Sorry for the long post, and thank you by advance for enlighten me !
Best Answer
Let's take $Q=[0:1]$, and always stick to the preimages of this point. Then the preimages only consist of two points $$[0: a], [a:a]\text{ where }a\not=0.$$
Near $Q\in C_2$, we can take the local neighborhood and take the uniformizer to be $t = x/y$. By the description of preimage points (non of the $y$-coordicate is 0), we could use the (same-expression) local parameter $u = x/y$ for any $P\in \phi^{-1}(Q)$. Then the pullback of $t$ is: $$\phi^*(t) = \dfrac{X^3(X-Y)^2}{Y^5} = u^3(u-1)^2.$$ This vanishes when $u=0$ or $u=1$. Thus when $u=0$, corresponding to the preimage point $[0:a]=[0:1]$, the ramification index is 3; and when $u=1$, corresponding to the preimage point $[a:a]=[1:1]$, the ramification index is 2.
you need to give homogeneous polynomials...