Non-semistable Arakelov bound and hyperelliptic Szpiro ratio for function fields
HTML articles powered by AMS MathViewer
- by Khac Viet Nguyen
- Proc. Amer. Math. Soc. 127 (1999), 3125-3130
- DOI: https://doi.org/10.1090/S0002-9939-99-05426-X
- Published electronically: July 12, 1999
- PDF | Request permission
Abstract:
We prove a complex function field analogue of Szpiro’s conjecture for hyperelliptic curves and some applications. The cases of function fields of positive characteristic and number fields are discussed briefly.References
- S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269–1293 (Russian). MR 0321933
- Dorian Goldfeld and Lucien Szpiro, Bounds for the order of the Tate-Shafarevich group, Compositio Math. 97 (1995), no. 1-2, 71–87. Special issue in honour of Frans Oort. MR 1355118
- M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), no. 2, 419–450. MR 948108, DOI 10.1007/BF01394340
- P. Lockhart, On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc. 342 (1994), no. 2, 729–752. MR 1195511, DOI 10.1090/S0002-9947-1994-1195511-X
- Shigeru Matsusaka, Some numerical invariants of hyperelliptic fibrations, J. Math. Kyoto Univ. 30 (1990), no. 1, 33–57. MR 1041711, DOI 10.1215/kjm/1250520121
- Atsushi Moriwaki, Bogomolov conjecture over function fields for stable curves with only irreducible fibers, Compositio Math. 105 (1997), no. 2, 125–140. MR 1440719, DOI 10.1023/A:1000139117766
- Nguyen Khac Viet, A complete proof of Beauville’s conjecture, Tạp chí Toán Học J. Math. 22 (1994), no. 3-4, 114–116. MR 1368119
- Nguyen, K. V., On families of curves over $\mathbb {P}^{1}$ with small number of singular fibres, C. R. Acad. Sci. Paris, 326 (1998), Série I, 459–463.
- Tetsuji Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24 (1972), 20–59. MR 429918, DOI 10.2969/jmsj/02410020
- Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, Société Mathématique de France, Paris, 1981 (French). Astérisque No. 86 (1981). MR 642675
- Kenji Ueno, Discriminants of curves of genus $2$ and arithmetic surfaces, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 749–770. MR 977781
Bibliographic Information
- Khac Viet Nguyen
- Affiliation: Institute of Mathematics, P.O. Box 631, Bo Ho, 10000, Hanoi, Vietnam
- Email: nkviet@thevinh.ncst.ac.vn
- Received by editor(s): June 12, 1997
- Published electronically: July 12, 1999
- Additional Notes: The research was partially supported by the National Natural Science Basic Research Foundation of Vietnam.
- Communicated by: Ron Donagi
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 3125-3130
- MSC (1991): Primary 11G30, 14H05
- DOI: https://doi.org/10.1090/S0002-9939-99-05426-X
- MathSciNet review: 1676300
Dedicated: Dedicated to the memory of Professor K. Kodaira