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Unformatted text preview: ON THE INDICES OF CURVES OVER LOCAL FIELDS PETE L. CLARK Abstract. Fix a nonnegative 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 finitetype Kscheme 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 finitetype kscheme 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 Krational 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 Lrational 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 Qrational 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 Krational zerocycle 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 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 minvariant m ( V ) in [3]). 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
 Clark
 Algebra

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