analog for the discriminant of number fields in the function field case? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:00:48Z http://mathoverflow.net/feeds/question/95638 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95638/analog-for-the-discriminant-of-number-fields-in-the-function-field-case analog for the discriminant of number fields in the function field case? oxeimon 2012-05-01T06:08:48Z 2012-05-01T06:08:48Z <p>Is there a nice algebraic way of determining the ramification of a morphism between curves? Ie, some analog of the discriminant of number fields?</p> <p>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.</p>