Structure of the Mordell-Weil group over the $\mathbb {Z}_p$-extensions
HTML articles powered by AMS MathViewer
- by Jaehoon Lee PDF
- Trans. Amer. Math. Soc. 373 (2020), 2399-2425 Request permission
Abstract:
We study the $\Lambda$-module structure of the Mordell-Weil, Selmer, and Tate-Shafarevich groups of an abelian variety over $\mathbb {Z}_p$-extensions.References
- J. Coates and R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), no. 1-3, 129–174. MR 1369413, DOI 10.1007/s002220050048
- Henri Darmon, Fred Diamond, and Richard Taylor, Fermat’s last theorem, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 1–154. MR 1474977
- Matthias Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990), 113–127. MR 1079004, DOI 10.1515/crll.1990.412.113
- Ralph Greenberg, Iwasawa theory for $p$-adic representations, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97–137. MR 1097613, DOI 10.2969/aspm/01710097
- Ralph Greenberg, Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) Lecture Notes in Math., vol. 1716, Springer, Berlin, 1999, pp. 51–144. MR 1754686, DOI 10.1007/BFb0093453
- Ralph Greenberg, Introduction to Iwasawa theory for elliptic curves, Arithmetic algebraic geometry (Park City, UT, 1999) IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 407–464. MR 1860044, DOI 10.1090/pcms/009/06
- Hideo Imai, A remark on the rational points of abelian varieties with values in cyclotomic $Z_{p}$-extensions, Proc. Japan Acad. 51 (1975), 12–16. MR 371902
- Somnath Jha and Dipramit Majumdar, Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms, Asian J. Math. 21 (2017), no. 3, 397–428. MR 3672212, DOI 10.4310/AJM.2017.v21.n3.a1
- Somnath Jha and Aprameyo Pal, Algebraic functional equation for Hida family, Int. J. Number Theory 10 (2014), no. 7, 1649–1674. MR 3256845, DOI 10.1142/S1793042114500493
- Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
- Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
- J. S. Milne, Arithmetic duality theorems, 2nd ed., BookSurge, LLC, Charleston, SC, 2006. MR 2261462
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
- Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
- Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
Additional Information
- Jaehoon Lee
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
- Address at time of publication: Math department of KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141 South Korea
- MR Author ID: 1101046
- Email: jaehoonlee@kaist.ac.kr
- Received by editor(s): September 24, 2018
- Received by editor(s) in revised form: May 25, 2019, and June 12, 2019
- Published electronically: December 20, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 2399-2425
- MSC (2010): Primary 11R23, 11Gxx
- DOI: https://doi.org/10.1090/tran/7937
- MathSciNet review: 4069223