Structure of the Mordell-Weil group over the ℤ_{𝕡}-extensions

Lee, Jaehoon

doi:10.1090/tran/7937

Structure of the Mordell-Weil group over the $\mathbb {Z}_p$-extensions
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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