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Euler characteristics and their congruences in the positive rank setting

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  11 June 2020

Anwesh Ray  and
R. Sujatha
Anwesh Ray*
Affiliation:
Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY14853-4201, USA
R. Sujatha
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada e-mail: sujatha@math.ubc.ca
*
e-mail: ar2222@cornell.edu
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Abstract

The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Keywords

Iwasawa theory of elliptic curves

MSC classification

Primary: 11R23: Iwasawa theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 228 - 245
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Copyright
© Canadian Mathematical Society 2020

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References

Balakrishnan, J., Müller, J. S., and Stein, W., A $\;p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties. Math. Comp. 85(2016), no. 298, 983–1016. https://doi.org/10.1090/mcom/3029CrossRefGoogle Scholar
Coates, J. H. and Greenberg, R., Kummer theory for abelian varieties over local fields. Invent. Math. 124(1996), 129–174. https://doi.org/10.1007/s002220050048CrossRefGoogle Scholar
Coates, J. H. and Howson, S., Euler characteristics and elliptic curves II. J. Math. Soc. Jpn. 53(2001), 175–235. https://doi.org/10.2969/jmsj/05310175CrossRefGoogle Scholar
Coates, J. H., Schneider, P., and Sujatha, R., Links between cyclotomic and $\textrm{GL}_2$-Iwasawa theory. Doc. Math. Extra Volume(2003), 187–215.CrossRefGoogle Scholar
Coates, J. H. and Sujatha, R., Galois cohomology of elliptic curves. Tata Institute of Fundamental Research Lectures on Mathematics, 88, Narosa House, New Delhi, 2000.Google Scholar
Emerton, M., Pollack, R., and Weston, T., Variation of Iwasawa invariants in Hida families. Invent. Math. 163(2006), no. 3, 523–580. https://doi.org/10.1007/s00222-005-0467-7CrossRefGoogle Scholar
Greenberg, R., Iwasawa theory for elliptic curves. In: Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999, 51–144. https://doi.org/10.1007/BFb0093453CrossRefGoogle Scholar
Greenberg, R. and Vatsal, V., On the Iwasawa invariants of elliptic curves. Invent. Math. 142(2000), 17–63. https://doi.org/10.1007/s002220000080Google Scholar
Hachimori, Y. and Venjakob, O., Completely faithful Selmer groups over Kummer extensions. Doc. Math. Extra Volume(2003), 443–478.Google Scholar
Kato, K., p-adic Hodge theory and values of zeta functions of modular forms. Astérisque 295(2004), 117–290.Google Scholar
Lim, M. F. and Sujatha, R., Fine Selmer groups of congruent Galois representations. J. Number Theory 187(2018), 66–91. https://doi.org/10.1016/j.jnt.2017.10.018Google Scholar
Mazur, B. and Swinnerton-Dyer, P., Arithmetic of Weil curves. Invent. Math. 25(1974), 1–61. https://doi.org/10.1007/BF01389997CrossRefGoogle Scholar
Mazur, B., Tate, J., and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84(1986), 1–48. https://doi.org/10.1007/BF01388731CrossRefGoogle Scholar
Perrin-Riou, B., Théorie d’Iwasawa et hauteurs p-adiques. Invent. Math. 109(1992), 137–185. https://doi.org/10.1007/BF01232022Google Scholar
Silverman, J. H., The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106, 2nd edition, Springer, Dordrecht, 2009. https://doi.org/10.1007/978-0-387-09494-6Google Scholar
Schneider, P., p-adic height pairings. I. Invent. Math. 69(1982), 401–409. https://doi.org/10.1007/BF01389362CrossRefGoogle Scholar
Schneider, P., p-adic height pairings. II. Invent. Math. 79(1985), 329–374. https://doi.org/10.1007/BF01388978CrossRefGoogle Scholar
Serre, J. P., Abelian l-adic representations and elliptic curves. Research Notes in Mathematics 7, AK Peters, Wellesley, MA, 1998.Google Scholar
Shekhar, S. and Sujatha, R., Euler characteristic and congruences of elliptic curves. Münster J. Math. 7(2014), 327–343.Google Scholar
Skinner, C. and Urban, E., The Iwasawa main conjectures for ${\textrm{GL}}_2$. Invent. Math. 195(2014), 1–277. https://doi.org/10.1007/s00222-013-0448-1CrossRefGoogle Scholar
Vatsal, V., Canonical periods and congruence formulae. Duke Math. J. 98(1999), 397–419. https://doi.org/10.1215/S0012-7094-99-09811-3CrossRefGoogle Scholar
Zerbes, S. L., Generalised Euler characteristics of Selmer groups. Proc. London Math. Society 98(2009), 775–796. https://doi.org/10.1112/plms/pdn049CrossRefGoogle Scholar