-2
$\begingroup$

Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$. The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ through the action on the coefficients of the equations defining $X$.

So, if for example $X$ is defined in $\mathbb{P}^2$ by the equation $\{F=0\}$ then $\sigma\in Gal(\mathbb{C},\mathbb{Q})$ sends $\{F=0\}$ to $\{F^\sigma=0\}$ and this action corresponds to the action of $\sigma$ on the coordinates of the points ($P\mapsto P^\sigma$).

Now let's take $f\in \mathcal{M}(X)$ i.e. a meromorphic function on $X$. If $P\in X$ is a zero for $f$, i want to prove that $ord_P(f)=ord_{P^\sigma}(f^\sigma)$.

If $X=\mathbb{P}^1$ this is straightforward, but i don't know how to deal with two ore more homogeneous variables (i think i'm forced to see $X$ as an algebraic curve in $\mathbb{P}^n$ to see the action of $Gal(\mathbb{C},\mathbb{Q})$, so $f$ is a rational function in the homogeneous coordinates of $\mathbb{P}^n$).

$\endgroup$
4
  • $\begingroup$ It seems to me that, if you replace "meromorphic" with "rational" (which they are), then everything is defined in purely algebraic terms (like the maximal power of the maximal ideal that contains the germ) and hence is obviously invariant under the Galois action. $\endgroup$ Mar 3, 2014 at 21:35
  • $\begingroup$ The map $X\to X^\sigma$ is an isomorphism of schemes (though not of $\mathbb{C}$-schemes). The order of vanishing of a function at a closed point does not depend on the structural morphism $\to Spec(\mathbb{C}$. $\endgroup$ Mar 3, 2014 at 21:36
  • $\begingroup$ To Piotr Achinger: sorry, but i'd prefer not to use the language of the schemes.. To Alex Degtyarev: yes, but the point i'm missing is exactly why it is "obviously invariant under the Galois action" $\endgroup$ Mar 3, 2014 at 22:21
  • $\begingroup$ Because everything you can construct using just algebra is invariant under the Galois action, because the Galois group preserves addition, multiplication, subtraction, and division. $\endgroup$
    – Will Sawin
    Mar 4, 2014 at 5:24

1 Answer 1

1
$\begingroup$

This question amounts to tracing the ideal-theoretic interpretations of all of the objects in question. You have a coordinate ring $S$ for projective space, and a homogeneous ideal $I$ defining $X$. Complex conjugation takes $I$ to $I^\sigma$ (i.e., the coefficients of all functions are conjugated), and the homogeneous coordinate ring $S/I$ of $X$ is taken to the homogeneous coordinate ring $S^\sigma/I^\sigma$ of $X^\sigma$. Points in $X$ are in bijection with maximal relevant ideals $J$ containing $I$, and complex conjugation respects containment.

Any point $P \in X$ has a local ring $R$ composed of the meromorphic functions on $X$ that don't have a pole at $P$ (it is made by localizing $S/I$), and the maximal ideal $m$ is made from those functions that vanish at $P$. The order of vanishing of a meromorphic function $f$ at $P$ is the supremum of integers $k$ such that $f \in m^k$. For your question, it suffices to note that complex conjugation takes $R \to R^\sigma$ and hence $m$ to $m^\sigma$. In particular, $f \in m^k$ if and only if $f^\sigma \in (m^\sigma)^k$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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