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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symmetric roots and admissible pairing


Author: Robin de Jong
Journal: Trans. Amer. Math. Soc. 363 (2011), 4263-4283
MSC (2010): Primary 11G20, 14G40
Published electronically: March 3, 2011
MathSciNet review: 2792987
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Abstract: Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant $ \chi$. We provide a simple geometric interpretation for this invariant $ \chi$, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant $ \chi$ coincides with the invariant $ \varphi$ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus $ 2$ case, and we obtain a new proof of the Bogomolov conjecture for curves of genus $ 2$ over number fields.


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Additional Information

Robin de Jong
Affiliation: Mathematical Institute, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: rdejong@math.leidenuniv.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05217-7
PII: S 0002-9947(2011)05217-7
Keywords: Hyperelliptic curves, local fields, admissible pairing, self-intersection of the relative dualising sheaf, symmetric roots
Received by editor(s): June 29, 2009
Received by editor(s) in revised form: October 5, 2009
Published electronically: March 3, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.