2
$\begingroup$

I am reading from "Topics in Galois Theory" by Serre.

enter image description here

I want to study the proof of this lemma. He has not provided the proof. Kindly mention some reference.

$\endgroup$
3
$\begingroup$

This follows from the Riemann-Hurwitz formula. Namely, if you have a finite degree $N$ map of curves $Y\rightarrow X$ over a number field of genus $g_Y,g_X$ respectively, then we have: $$2g_Y-2 = N(2g_X-2) + \sum_{P\in Y}(e_P-1)$$ where $e_P$ is the ramification index at $P$.

You can find this formula all over the place. In your case, $X = \mathbb{P}^1_k$ with function field $k(T)$. If $H$ is index 2 in $G$, then the fixed field of $H$ is the function field of a curve $Y$ of degree 2 over $X$, ramified at most above 3 points. Since the map is degree 2, the ramification indices are all either 1 or 2, and so you get: $$2g_Y - 2 = 2\cdot(-2) + (0\text{ or }1) + (0\text{ or }1) + (0\text{ or }1)$$ since $g_Y$ is a nonnegative integer, this shows that in fact $Y$ is actually only ramified over 2 points, and $g_Y = 0$.

Note that if you allow ramification over 4 points, then you can get a genus 1 degree 2 cover - namely, the quotient of an elliptic curve by its involution $[-1]$.

EDIT: If $K$ is a field, then a place of $K$ is by definition an equivalence class of absolute values on $K$, where two absolute values are equivalent if they define the same (metric) topology on $K$. An easy starting point is:

https://en.wikipedia.org/wiki/Algebraic_number_field#Places

If $X$ is a curve over a field $k$, then locally it is the Spec of a $k$-algebra $A$, e.g. $A = k[x]$, or $A = k[x,y]/(f)$...etc. A $k$-rational point is just a maximal ideal $m$ with residue field $k$. Given such a maximal ideal, the localization $A_m$ is a discrete valuation ring (wikipedia this), which naturally comes with a "valuation map" $v_m : A_m\rightarrow\mathbb{N}$, which extends to a valuation $$K := \text{Frac }A_m\stackrel{v}{\longrightarrow}\mathbb{Z}$$ The properties of this valuation ensure that if we define: $$|x| := 2^{-v_m(x)}\qquad\forall x\in K$$ then $|\cdot|$ is a (nonarchimedean) "absolute value" on $K$, where "2" is an arbitrary constant, and can be chosen to be any real number $> 1$ without affecting the equivalence class of the corresponding absolute value. Here, $K$ is precisely the function field of $X$, and hence every $k$-point $m$ defines a "$k$-rational place" of $K$, and different maximal ideals/points yield inequivalent places.

In a sentence, there is a bijection between $k$-rational points of a projective algebraic curve $X$ and $k$-rational places of its function field. Here, projective basically means "complete", in the sense that there are no points missing. If $X$ is an affine curve, you can always consider its "completion" by adding in the missing points. Note that however the function field of a projective curve is the same as the function field of any open subset, and hence there may be places of $\text{Frac}(A)$ which do not come from a maximal ideal of $A$. This is the connection between places and points.

Anyway, some good books to start with might include:

  1. Galois Groups and Fundamental Groups - Tamas Szamuely.
  2. Groups as Galois Groups - Volklein.
$\endgroup$
5
  • $\begingroup$ I think I don't have enough background to understand this. I am reading this book on my own. If you can clarify some points, it will be helpful for me. What does he mean by the following "places which are rational over $k$" and "..fixed field is rational". I have never see this terminology before. can you please explain . $\endgroup$ – Tensor_Product Nov 14 '16 at 20:29
  • $\begingroup$ I am 100% sure that your proof is correct, But I am not able to understand it completely. Can you mention some reference, from where I can read about this ramification thing. I also want to understand How we concluded that fixed field of $H$ is the function field of a curve $Y$ of degree $2$ over $X$. $\endgroup$ – Tensor_Product Nov 14 '16 at 20:36
  • $\begingroup$ I have lot of gaps to fill. Serre's book assumes that reader know about ramification, etc. He has not mentioned any source/reference to read the basic stuffs. It will helpful if you can mention some. $\endgroup$ – Tensor_Product Nov 14 '16 at 20:36
  • 1
    $\begingroup$ @Sri Added some additional stuff in an edit. Have fun! $\endgroup$ – Will Chen Nov 14 '16 at 20:54
  • 1
    $\begingroup$ @Sri Also, locally (in the complex topology) near any point $x$ in the domain, the map of curves in a small neighborhood of this point looks like the map $\mathbb{C}\rightarrow\mathbb{C}$ given by $z\mapsto z^n$ for some integer $n$. In this case we can define the ramification index to be $n$. $\endgroup$ – Will Chen Nov 14 '16 at 21:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.