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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An explicit theory of heights

Author: E. V. Flynn
Journal: Trans. Amer. Math. Soc. 347 (1995), 3003-3015
MSC: Primary 11G10; Secondary 11G30, 14H25, 14K15
MathSciNet review: 1297525
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Abstract: We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus $ > 1$, it is impractical to apply Hilbert's Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrary curve of genus $ 2$, and we apply the technique to compute generators of $ \mathcal{J}(\mathbb{Q})$, the Mordell-Weil group for a selection of rank $ 1$ examples.

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  • [1] J. Bost and J.-F. Mestre, Moyenne arithmético-géometrique et périodes des courbes de genre $ 1$ et $ 2$, Gaz. Math. 38 (1988), 36-64.
  • [2] J. W. S. Cassels, The Mordell-Weil group of curves of genus 2, Arithmetic and geometry papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, Vol. 1 Arithmetic, Birkhäuser, Boston, 1983, pp. 29-60. MR 717589 (84k:14032)
  • [3] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151 (93m:11053)
  • [4] E. V. Flynn, The Jacobian and formal group of a curve of genus $ 2$ over an arbitrary ground field, Math. Proc. Cambridge Philos. Soc. 107 (1990), 425-441. MR 1041476 (91b:14025)
  • [5] -, The group law on the Jacobian of a curve of genus 2, J. Reine Angew. Math. 438 (1993), 45-69. MR 1219694 (95b:14022)
  • [6] -, Descent via isogeny on the Jacobian of a curve of genus 2, Acta Arith. 66 (1994), 23-43. MR 1262651 (95g:11057)
  • [7] D. M. Gordon and D. Grant, Computing the Mordell-Weil rank of Jacobians of curves of genus 2, Trans. Amer. Math. Soc. 337 (1993), 807-824. MR 1094558 (93h:11057)
  • [8] R. W. H. T. Hudson, Kummer's quartic surface, Cambridge Univ. Press, Cambridge, 1905; reprint, 1990. MR 1097176 (92e:14033)
  • [9] D. Masser and G. Wüstholz, Fields of large transcendence degree generated by the values of elliptic functions, Invent. Math. 72 (1983), 407-464. MR 704399 (85g:11060)
  • [10] D. Mumford, Tata lectures on theta. I, II, Progr. in Math., vols. 28 and 43, Birkhäuser, Boston, 1983. MR 688651 (85h:14026)
  • [11] E. F. Schaefer, $ 2$-descent on the Jacobians of hyperelliptic curves, J. Number Theory (to appear). MR 1326746 (96c:11066)
  • [12] J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1986. MR 817210 (87g:11070)

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Article copyright: © Copyright 1995 American Mathematical Society

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