Unicritical rational functions on curves in characteristic $p$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:58:15Z http://mathoverflow.net/feeds/question/101332 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101332/unicritical-rational-functions-on-curves-in-characteristic-p Unicritical rational functions on curves in characteristic $p$ Xander Faber 2012-07-04T18:41:06Z 2012-07-04T18:41:06Z <p>Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$. </p> <blockquote> <p>How precisely can one describe the following set of positive integers: <code>$$D(X, x_0) = \{ d : \exists \text{ a (separable) morphism f : X \to \mathbb{P}^1 of degree d ramified only at x_0}\}?$$</code></p> </blockquote> <p>An equivalent formulation in terms of divisors is given by <code>$$D(X, x_0) = \{ d : \text{there exists a } g^2_d \text{ on X that is ramified only at x_0}\}.$$</code></p> <p>If $X = \mathbb{P}^1$, then I have a <a href="http://arxiv.org/abs/1102.1433" rel="nofollow">preprint on unicritical rational functions</a> that proves <code>$$D( \mathbb{P}^1, x_0 ) = \{ d &gt; 1: d \equiv 0 \text{ or } 1 \pmod{p} \}.$$</code> The proof uses continued fraction expansions of rational functions, and so it does not seem to generalize to higher genus curves. But the answer is so restricted that I became curious what might be true for an arbitrary curve $X$. </p> <p>Here is a related weaker question. </p> <blockquote> <p>Fix an integer $d > 1$. For which curves $X_{/k}$ and points $x_0 \in X(k)$ is it true that $d \in D(X, x_0)$? </p> </blockquote> <p>For example, consider $d=2$. The Riemann-Hurwitz formula shows that $x_0$ must be wildly ramified for any unicritical degree-$2$ morphism $X \to \mathbb{P}^1$. Hence $2 \not\in D(X, x_0)$, except perhaps when $k$ has characteristic~2. Moreover, one can construct hyperelliptic curves of any genus in characteristic $2$ ramified at only a single point over $\mathbb{P}^1$, which must necessarily be a Weierstrass point. This shows that $D(X, x_0)$ is likely to be very sensitive to the choice of point $x_0$ when $X$ has genus $g > 1$. (The automorphism groups of genus-0 and genus-1 curves act transitively on $k$-rational points, so $D(X, x_0)$ is independent of the choice of point $x_0$ in these cases.) </p>