Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T16:04:59.210Z Has data issue: false hasContentIssue false
  • English
  • Français

The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

Published online by Cambridge University Press:  20 November 2018

Fumio Sairaiji  and
Takuya Yamauchi
Fumio Sairaiji
Affiliation:
Hiroshima International University, Hiro, Hiroshima 737-0112, Japane-mail: [email protected]
Takuya Yamauchi
Affiliation:
Faculty of Education, Kagoshima University, 1-20-6 Korimoto, Kagoshima, 890-0065, Japane-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell–Weil rank over the maximal abelian extension ${{K}^{\text{ab}}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C\left( {{K}^{\text{ab}}} \right)\,=\,\infty $ and for any abelian variety of $\text{G}{{\text{L}}_{2}}$-type with trivial character.

Keywords

11G0511D2514G2514K07Mordell–Weil rankJacobian varietiesFrey-Jarden conjectureabelian points
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 842 - 849
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Baker, M. and Poonen, B., Torsion packets on curves. Compositio Math. 127(2001), no. 1, 109–116. http://dx.doi.org/10.1023/A:1017557230785 Google Scholar
[2] Billing, G., Vom Range kubischer Kurven vom Geschlecht Eins in algebraischen Rationalitatsbereichen. Yhdeksas skandinaavinen matemaatikkokongressi. Helsingissa 23-26. elokuuta 1938, pp. 146–150.Google Scholar
[3] Faltings, G., Endlichkeitssätze für abelsche Varietäten/”uber Zahlkörpern. Invent. Math. 73(1983), no. 3, 349–366. http://dx.doi.org/10.1007/BF01388432 Google Scholar
[4] Frey, G. and Jarden, M., Approximation theory and the rank of abelian varieties over large algebraic fields. Proc. London Math. Soc. 28(1974), 112–128. http://dx.doi.org/10.1112/plms/s3-28.1.112 Google Scholar
[5] Imai, H., On the rational points of some Jacobian varieties over large algebraic number fields. Kodai Math. J. 3(1980), no. 1, 56–58. http://dx.doi.org/10.2996/kmj/1138036119 Google Scholar
[6] Khare, C. and Wintenberger, J-P., Serre's modularity conjecture. I, II. Invent. Math. 178(2009), no. 3, 485–504, 505–586.Google Scholar
[7] Kisin, M., Moduli of finite flat group schemes and modularity. Ann. of Math. 170(2009), no. 3, 1085–1180. http://dx.doi.org/10.4007/annals.2009.170.1085 Google Scholar
[8] Kisin, M., Modularity of 2-adic Barsotti-Tate representations. Invent. Math. 178(2009), no. 3, 587–634. http://dx.doi.org/10.1007/s00222-009-0207-5 Google Scholar
[9] Kobayashi, E., A remark on the Mordell-Weil rank of elliptic curves over the maximal abelian extension of the rational number field. Tokyo J. Math. 29(2006), no. 2, 295–300. http://dx.doi.org/10.3836/tjm/1170348168 Google Scholar
[10] Moon, H., On the Mordell-Weil groups of Jacobians of hyperelliptic curves over certain elementary abelian 2-extensions. Kyungpook Math. J. 49(2009), no. 3, 419–424.Google Scholar
[11] Murabayashi, N., Mordell-Weil rank of the Jacobians of the curves defined by yp = f(x). Acta Arith 64(1993), no. 4, 297–302.Google Scholar
[12] Murty, K. and Murty, R., Mean values of derivatives of modular L-series. Ann. of Math. 133(1991), no. 3, 447–475. http://dx.doi.org/10.2307/2944316 Google Scholar
[13] Néron, A., Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps. Bull. Soc. Math. France 80(1952), 101–166.Google Scholar
[14] Neukirch, J., Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften 322. Springer-Verlag, Berlin, 1999.Google Scholar
[15] Petersen, S., On a question of Frey and Jarden about the rank of abelian varieties. J. Number Theory 120(2006), no. 2, 287–302. http://dx.doi.org/10.1016/j.jnt.2005.12.006 Google Scholar
[16] Pop, F., Embedding problems over large fields. Ann. of Math. 144(1996), no. 1, 1–34. http://dx.doi.org/10.2307/2118581 Google Scholar
[17] Raynaud, M., Courbes sur une variété abélienne et points de torsion. Invent. Math. 71(1983), no. 1, 207–233. http://dx.doi.org/10.1007/BF01393342 Google Scholar
[18] Ribet, K., Abelian varieties over ℚ and modular forms. In: Modular Curves and Abelian Varieties, 241–261, Progr. Math. 224. Birkhäuser, Basel, 2004.Google Scholar
[19] Ribet, K., Torsion points of abelian varieties in cyclotomic extensions. Appendix to Katz, N. M. and Lang, S., Finiteness theorems in geometric classified theory. Enseign. Math. 27(1982), no. 3–4, 285–319.Google Scholar
[20] Rosen, M. and Wong, S., The rank of abelian varieties over infinite Galois extensions. J. Number Theory 92(2002), no. 1, 182–196. http://dx.doi.org/10.1006/jnth.2001.2692 Google Scholar
[21] Ruppert, W. M., Torsion points of abelian varieties over abelian extensions. arxiv:math/9803169Google Scholar
[22] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Publications of the Mathematical Society of Japan 11. Iwanami Shoten Publishers, Tokyo, Princeton University Press. 1971.Google Scholar
[23] Top, J., A remark on the rank of Jacobians of hyperelliptic curves over ℚ over certain elementary abelian 2-extensions. Tohoku. Math. J. 40(1988), no. 4, 613–616. http://dx.doi.org/10.2748/tmj/1178227925 Google Scholar
[24] Zarhin, Y. G., Endomorphisms and torsion of abelian varieties. Duke. Math. J. 54(1987), no. 1, 131–145. http://dx.doi.org/10.1215/S0012-7094-87-05410-X Google Scholar
[25] Zhang, S.-W., Elliptic curves, L-functions, and CM-points. In: Current Developments in Mathematics, 2001. Int. Press, Somerville, MA, 2002, pp. 179–219.Google Scholar