Is there a nice algebraic way of determining the ramification of a morphism between curves? Ie, some analog of the discriminant of number fields?

Specifically, I'm trying to prove that if $X$ is a curve over $\mathbb{C}$ that's defined over $\overline{\mathbb{Q}}$, and $t : X\rightarrow\mathbb{P}^1_\mathbb{C}$ is a morphism also defined over $\overline{\mathbb{Q}}$, then the critical values of $t$ (ie, the points of $\mathbb{P}^1_\mathbb{C}$ above which $t$ is ramified) are $\overline{\mathbb{Q}}$-rational.

`$\mathbb{P}^1_{\mathbb{C}}$`

This is the (slightly more global) function field analogue of the co-different ideal (that is, the inverse of the different). It's just the co-kernel of the map`$t^*:\Omega^1_{\mathbb{P}^1}\to\Omega^1_X$`

. If $t$ is defined over $\overline{\mathbb{Q}}$, then so is its relative sheaf of differentials, and so also is its support. – Keerthi Madapusi Pera May 1 '12 at 7:00