Iwasawa theory of the fine Selmer group
Author:
Christian Wuthrich
Journal:
J. Algebraic Geom. 16 (2007), 83-108
DOI:
https://doi.org/10.1090/S1056-3911-06-00436-X
Published electronically:
June 21, 2006
MathSciNet review:
2257321
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Abstract |
References |
Additional Information
Abstract: The fine Selmer group of an elliptic curve $E$ over a number field $K$ is obtained as a subgroup of the usual Selmer group by imposing stronger conditions at places above $p$. We prove a formula for the Euler-characteristic of the fine Selmer group over a $\mathbb {Z}_p$-extension and use it to compute explicit examples.
References
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- Bernadette Perrin-Riou, Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque 229 (1995), 198 (French, with English and French summaries). MR 1327803
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- Christian Wuthrich, The fine Selmer group and height pairings, Ph.D. thesis, University of Cambridge, UK, 2004.
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- ---, The fine Tate-Shafarevich group, to appear in Math. Proc. Cambridge Philos. Soc., 2005.
References
- Dominique Bernardi, Hauteur $p$-adique sur les courbes elliptiques, Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser Boston, 1981, pp. 1–14. MR 0633886 (83i:14030)
- Dominique Bernardi and Bernadette Perrin-Riou, Variante $p$-adique de la conjecture de Birch et Swinnerton-Dyer (le cas supersingulier), C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 3, 227–232. MR 1233417 (94k:11071)
- John Coates, Ralph Greenberg, Kenneth A. Ribet, and Karl Rubin, Arithmetic theory of elliptic curves, Lecture Notes in Mathematics, vol. 1716, Springer, 1999, Lectures from the 3rd C.I.M.E. Session held in Cetraro, July 12–19, 1997. MR 1754684 (2000m:11002)
- John Coates and Ramdorai Sujatha, Galois cohomology of elliptic curves, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 88, Narosa Publishing House, 2000. MR 1759312 (2001b:11046)
- ---, Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, Math. Ann. 331 (2005), no. 4, 809–839. MR 2148798
- Hideo Imai, A remark on the rational points of abelian varieties with values in cyclotomic $\mathbb {Z}_{p}$-extensions, Proc. Japan Acad. 51 (1975), 12–16. MR 0371902 (51:8119)
- Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Cohomologies $p$-adiques et application arithmétiques. III, Astérisque, vol. 295, Société Mathématique de France, Paris, 2004. MR 2104361 (2006b:11051)
- Shin-ichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1–36. MR 1965358 (2004b:11153)
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Springer, 2000. MR 1737196 (2000j:11168)
- Bernadette Perrin-Riou, Théorie d’Iwasawa et hauteurs $p$-adiques, Invent. Math. 109 (1992), no. 1, 137–185. MR 1168369 (93g:11109)
- ---, Fonctions $L$ $p$-adiques d’une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 945–995. MR 1252935 (95d:11081)
- ---, Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque (1995), no. 229, 198. MR 1327803 (96e:11062)
- Robert Pollack, Iwasawa invariants of elliptic curves, Available online at the address http://math.bu.edu/people/rpollack/Data/data.html.
- ---, On the $p$-adic $L$-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523–558. MR 1983040 (2004e:11050)
- Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000, Hermann Weyl Lectures. The Institute for Advanced Study. MR 1749177 (2001g:11170)
- Peter Schneider, $p$-adic height pairings. II, Invent. Math. 79 (1985), no. 2, 329–374. MR 0778132 (86j:11063)
- Christian Wuthrich, The fine Selmer group and height pairings, Ph.D. thesis, University of Cambridge, UK, 2004.
- ---, On $p$-adic heights in families of elliptic curves, J. London Math. Soc. (2) 70 (2004), no. 1, 23–40. MR 2064750
- ---, The fine Tate-Shafarevich group, to appear in Math. Proc. Cambridge Philos. Soc., 2005.
Additional Information
Christian Wuthrich
Affiliation:
Section de mathématiques, CSAG, École polytechnique fédérale, 1015 Lausanne, Switzerland
MR Author ID:
681572
Email:
christian.wuthrich@epfl.ch
Received by editor(s):
May 22, 2005
Received by editor(s) in revised form:
October 7, 2005
Published electronically:
June 21, 2006
