On the indices of curves over local fields

Abstract

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. 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. Some terminological conventions: by a variety (resp. a curve) over a field 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 field 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) = ∅.1 Note that this holds -vacuously -even when g = 1.

On nondegeneracy of curves

“…As in Remark 4.2, an argument based on the index shows that there exist genus 4 curves that are not nondegenerate. A result by Clark [2007] states that for every g ≥ 2, there exists a number field k and a genus g curve V over k, such that the index of V is equal to 2g − 2, the degree of the canonical divisor.…”

On the Hasse principle for Shimura curves

“…When this paper was first written, I was optimistic that this condition might be superfluous: I did not know any family of semistable curves that violated it. I have since learned better: in [5] I construct, for every genus g = 1, a genus g curve over a local field with (the largest possible) m-invariant |2g − 2| by lifting curves over the residue field which are degenerate, with 2g − 2 rational components which are transitively permuted by Galois. This shows the importance of the word "split" in Lemma 20.…”

Period and index of genus one curves over global fields

“…For other results exhibiting curves with various periods and indices, see Cassels [1], Lang-Tate [8], Stein [20], O'Neil [14], and Clark [2], [3] and [4]. The strongest previous results in this direction are Clark's, who proved that when E[p] ⊂ E(K) for p prime, there are principal homogeneous spaces for E with period p and index p 2 , and that there are curves of every index over every number field.…”