Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2011-05217-7
Published electronically: March 3, 2011
MathSciNet review: 2792987
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. S. Bosch, Formelle Standardmodelle hyperelliptischer Kurven. Math. Ann. 251 (1980), no. 1, 19-42. MR 583822 (82b:14018)
  • 2. J.-B. Bost, J.-F. Mestre, L. Moret-Bailly, Sur le calcul explicite des ``classes de Chern'' des surfaces arithmétiques de genre $ 2$. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 69-105. MR 1065156 (92g:14018b)
  • 3. G. Faltings, Calculus on arithmetic surfaces. Ann. of Math. 119 (1984), no. 2, 387-424. MR 740897 (86e:14009)
  • 4. J. Guàrdia, Jacobian Nullwerte, periods and symmetric equations for hyperelliptic curves. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, 1253-1283. MR 2339331 (2008g:11105)
  • 5. R. de Jong, Explicit Mumford isomorphism for hyperelliptic curves. J. Pure Appl. Algebra 208 (2007), no. 1, 1-14. MR 2269824 (2007k:14046)
  • 6. R. de Jong, Admissible constants for genus 2 curves. Bulletin of the LMS 42 (2010), 405-411. MR 2651934
  • 7. I. Kausz, A discriminant and an upper bound for $ \omega^2$ for hyperelliptic arithmetic surfaces. Compositio Math. 115 (1999), no. 1, 37-69. MR 1671741 (2000e:14033)
  • 8. O.A. Laudal and K. Lønsted, Deformations of curves. I. Moduli for hyperelliptic curves. Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) 150-167, Lecture Notes in Mathematics 687, Springer, Berlin, 1978. MR 527233 (80g:14027)
  • 9. P. Lockhart, On the discriminant of a hyperelliptic curve. Trans. Amer. Math. Soc. 342 (1994), no. 2, 729-752. MR 1195511 (94f:11054)
  • 10. K. Lønsted, S.L. Kleiman, Basics on families of hyperelliptic curves. Compositio Math. 38 (1979), no. 1, 83-111. MR 523266 (80g:14028)
  • 11. S. Maugeais, Relèvement des revêtements $ p$-cycliques des courbes rationnelles semi-stables. Math. Ann. 327 (2003), no. 2, 365-393. MR 2015076 (2004j:14031)
  • 12. L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques. Invent. Math. 98 (1989), no. 3, 491-498. MR 1022303 (91h:14023)
  • 13. A. Moriwaki, Bogomolov conjecture for curves of genus $ 2$ over function fields. J. Math. Kyoto Univ. 36 (1996), no. 4, 687-695. MR 1443744 (98e:14029)
  • 14. Séminaire de géométrie algébrique du Bois Marie 1960-61 I: Revêtements étales et groupe fondamental. Dirigé par A. Grothendieck. Documents Mathématiques 3, Société Mathématique de France 2003. MR 2017446 (2004g:14017)
  • 15. K. Yamaki, Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic. Asian J. Math. 8 (2004), no. 3, 409-426. MR 2129243 (2005j:14039)
  • 16. K. Yamaki, Effective calculation of the geometric height and the Bogomolov conjecture for hyperelliptic curves over function fields. J. Math. Kyoto Univ. 48 (2008), no. 2, 401-443. MR 2436745 (2009h:14042)
  • 17. S.-W. Zhang, Admissible pairing on a curve. Invent. Math. 112 (1993), no. 1, 171-193. MR 1207481 (94h:14023)
  • 18. S.-W. Zhang, Gross-Schoen cycles and dualising sheaves. Invent. Math. 179 (2010), 1-73. MR 2563759

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G20, 14G40

Retrieve articles in all journals with MSC (2010): 11G20, 14G40


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: https://doi.org/10.1090/S0002-9947-2011-05217-7
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.

American Mathematical Society