Linear series over real and $p$-adic fields

Abstract

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve C of genus g to P 1 with prescribed ramification also yields weaker results when working over the real numbers or p-adic fields. Specifically, let k be such a field: we see that given g, d, n, and e 1 ,..e n satisfying Σ ι (e i -1) = 2d - 2 - g, there exists smooth curves C of genus g together with points P 1 ,..., P n such that all maps from C to P 1 can, up to automorphism of the image, be defined over k. We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case C = P 1 , n = 3, and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.