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A discriminant and an upper bound for w2 for hyperelliptic arithmetic surfaces

Published online by Cambridge University Press:  04 December 2007

IVAN KAUSZ
IVAN KAUSZ
Affiliation:
Mathematisches Institut der Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany. E-mail: kausz@mi.uni-koeln.de
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Abstract

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We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.

Keywords

hyperelliptic curvesarithmetic surfacesArakelov theory.
Type
Research Article
Information
Compositio Mathematica , Volume 115 , Issue 1 , January 1999 , pp. 37 - 69
Copyright
© 1999 Kluwer Academic Publishers