On the indices of curves over local fields

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I. We can in fact give a complete description of the finite extensions L/K such that C has an L-rational point. Applications are discussed to the corresponding problem over number fields. S. Sharif, in his 2006 Berkeley thesis, has independently obtained similar (but not identical) results. Our proof, however, is different: via deformation theory, we reduce to the problem of finding suitable actions of cyclic groups on finite graphs.