On the indices of curves over local fields

Details On the indices of curves over local fields On the indices of curves over local fields

2006-11-06

arxiv.org/abs/math/0611150v1

Mathematics

Number Theory

11 pages

Scientific paper

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.

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