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Unformatted text preview: ON THE INDICES OF CURVES OVER LOCAL FIELDS PETE L. CLARK Abstract. Fix a non-negative integer g and a positive integer I dividing 2 g- 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 g k = 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 dimen- sion 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 = ∅ . 1. Introduction Given a variety V defined over a field K , one would like to determine whether V has a K-rational point, and if it does not, to say something about S ( V ), the set of finite field extensions L/K for which V acquires an L-rational point. This is a very difficult problem: e.g., it is believed by many (but unproved) that there is no algorithm for the task of deciding whether a variety V / Q has a Q-rational point. In order to quantify the second part of the question, we introduce the index I ( V ) of a variety V /K : it is the greatest common divisor of all degrees of closed points on V . Equivalently, I ( V ) is the least positive degree of a K-rational zero-cycle on V . For a curve C /K , the index is equal to the least positive degree of a line bundle on C /K . If C has genus g , then the canonical bundle Ω 1 C/K has degree equal to 2 g- 2 and I ( C ) | 2 g- 2. 1 To know I ( V ) is much less than to know S ( V ): we need not know any partic- ular splitting field L/K , nor even the least possible degree of a splitting field (a quantity called the m-invariant m ( V ) in ). E.g. every variety over a finite field has I ( V ) = 1 (cf. Lemma 11); nevertheless computing S ( V ) is still a nontrivial task. 1 Note that this holds – vacuously – even when g = 1. 1 2 PETE L. CLARK It is natural to ask: Question 1. Fix a field K . For which pairs ( g,I ) ∈ N × Z + does there exist a curve C /K with I ( C ) = I ?...
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- Fall '10
- Algebra, Planar graph, Algebraic curve, pete l. clark, special ﬁber, arithmetic surface