Abstract
We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss a conjecture, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated \(\mathbb {Z}_p\)-module. The relationship between this conjecture and Iwasawa’s classical \(\mu =0\) conjecture is clarified. We also present some partial results towards the question whether the conjecture is invariant under isogenies.
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References
P. Billot, Quelques aspects de la descente sur une courbe elliptique dans le cas de réduction supersingulière, Compositio Math. 58 (1986), no. 3, 341–369.
A. Chandrakant, On the \(\mu \)-invariant of fine Selmer groups, J. Number Theory 135 (2014), 284–300.
J. Coates and R. Sujatha, Fine Selmer groups for elliptic curves with complex multiplication, Algebra and number theory, Hindustan Book Agency, Delhi, 2005, pp. 327–337.
T. Fukaya and K. Kato, A formulation of conjectures on \(p\)-adic zeta functions in non-commutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society (Providence, RI), vol. XII, Amer. Math. Soc. Transl. Ser. 2, no. 219, American Math. Soc., 2006, pp. 1–85.
R. Greenberg, The structure of Selmer groups, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11125–11128, Elliptic curves and modular forms (Washington, DC, 1996).
R. Greenberg, Iwasawa theory, projective modules, and modular representations, Mem. Amer. Math. Soc. 211 (2011), no. 992, vi+185.
S. Jha, Fine Selmer group of Hida deformations over non-commutative \(p\)-adic Lie extensions, Asian J. Math. 16 (2012), no. 2, 353–365.
S. Jha and R. Sujatha, On the Hida deformations of fine Selmer groups, J. Algebra 338 (2011), 180–196.
M. F. Lim, Notes on fine Selmer groups, preprint, arXiv:1306.2047v3, 2013.
J. S. Milne, Arithmetic duality theorems, second ed., BookSurge, LLC, Charleston, SC, 2006.
J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.
J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften, no. 323, Springer Verlag, Berlin Heidelberg, 2000.
J. L. Parish, Rational torsion in complex-multiplication elliptic curves, J. Number Theory 33 (1989), no. 2, 257–265.
B. Perrin-Riou, \(p\)-adic \(L\)-functions and \(p\)-adic representations, SMF/AMS Texts and Monographs, vol. 3, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2000, Translated from the 1995 French original by Leila Schneps and revised by the author.
P. Schneider, \(p\)-adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329–374.
J. H. Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009.
R. Sujatha, Elliptic curves and Iwasawa’s \(\mu =0\)conjecture, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. vol. 18, Springer, New York, 2010, pp. 125–135.
M. Witte, On a localisation sequence for the K-theory of skew power series rings, J. K-Theory 11 (2013), no. 1, 125–154.
M. Witte, On a noncommutative Iwasawa main conjecture for varieties over finite fields, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 2, 289–325.
C. Wuthrich, Iwasawa theory of the fine Selmer group, J. Algebraic Geom. 16 (2007), no. 1, 83–108.
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Sujatha, R., Witte, M. (2018). Fine Selmer Groups and Isogeny Invariance. In: Akbary, A., Gun, S. (eds) Geometry, Algebra, Number Theory, and Their Information Technology Applications. GANITA 2016. Springer Proceedings in Mathematics & Statistics, vol 251. Springer, Cham. https://doi.org/10.1007/978-3-319-97379-1_19
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DOI: https://doi.org/10.1007/978-3-319-97379-1_19
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