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On the discriminant of a hyperelliptic curve

Author: P. Lockhart
Journal: Trans. Amer. Math. Soc. 342 (1994), 729-752
MSC: Primary 11G30; Secondary 14H45
MathSciNet review: 1195511
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Abstract: The minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves. Over number fields this leads to a higher genus version of Szpiro's Conjecture. Analytically, the discriminant is shown to be related to Siegel modular forms of higher degree.

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  • [1] E. Arabello, et al., Geometry of algebraic curves, Grundlehren Math. Wiss. 267, Springer-Verlag, New York, 1985.
  • [2] Gerhard Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40. MR 853387
  • [3] Dorian Goldfeld, Modular elliptic curves and Diophantine problems, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 157–175. MR 1106659
  • [4] David Grant, A generalization of Jacobi’s derivative formula to dimension two, J. Reine Angew. Math. 392 (1988), 125–136. MR 965060,
  • [5] Jun-ichi Igusa, Theta functions, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194. MR 0325625
  • [6] Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
  • [7] P. Lockhart, Diophantine equations and the arithmetic of hyperelliptic curves, Thesis, Columbia University, 1990.
  • [8] Paul Lockhart, Michael Rosen, and Joseph H. Silverman, An upper bound for the conductor of an abelian variety, J. Algebraic Geom. 2 (1993), no. 4, 569–601. MR 1227469
  • [9] P. Lockhart, The ABC Conjecture implies Szpiro's Conjecture over arbitrary number fields, unpublished.
  • [10] David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651
  • [11] Joseph Oesterlé, Nouvelles approches du “théorème” de Fermat, Astérisque 161-162 (1988), Exp. No. 694, 4, 165–186 (1989) (French). Séminaire Bourbaki, Vol. 1987/88. MR 992208
  • [12] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • [13] J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math. 106, Springer-Verlag, New York, 1985.
  • [14] -, The abc-Conjecture implies Szpiro's Conjecture, unpublished.
  • [15] L. Szpiro, Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque No. 3 86 (1981), 44-78.
  • [16] Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451

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Keywords: Hyperelliptic, discriminant
Article copyright: © Copyright 1994 American Mathematical Society

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