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/K of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications are discussed to the corresponding problem over number fields. Notation and conventions Throughout this paper K shall denote a field and k a perfect field. We denote by k an algebraic closure of k and set gk = Gal(k/k), the absolute Galois group of k. From §2 onwards, K will be Henselian for a discrete valuation v, with valuation ring R and residue field k. By a variety (resp. a curve) over a field denoted K we will mean a finite-type K-scheme which is smooth, projective and geometrically integral (resp. of dimension one). By a variety (resp. a curve) over a perfect field denoted k we will mean a finite-type k-scheme which is geometrically integral (resp. of dimension one) but possibly incomplete or singular. If V is a variety defined over K and L/K is a field extension, we say that L splits V if V (L) 6= ∅.