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Growth of $\unicode[STIX]{x0428}$ in towers for isogenous curves

Part of: Diophantine equations Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  30 June 2015

Tim Dokchitser  and
Vladimir Dokchitser
Tim Dokchitser
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK email [email protected]
Vladimir Dokchitser
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email [email protected]
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Abstract

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

Keywords

elliptic curvesSelmer groupTate–Shafarevich groupp-adic Lie extensionsisogenyIwasawa theory𝜇-invariantabelian varieties

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05: Elliptic curves over global fields
Secondary: 11G10: Abelian varieties of dimension $> 1$ 11G07: Elliptic curves over local fields 11R23: Iwasawa theory 11D68: Rational numbers as sums of fractions
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 1981 - 2005
Copyright
© The Authors 2015 

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References

Betts, A. and Dokchitser, V., Finite quotients of $\mathbb{Z}[C_{n}]$-lattices and Tamagawa numbers of semistable abelian varieties, Preprint (2014), arXiv:1405.3151.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. I: The user language, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
Cassels, J. W. S., Arithmetic on curves of genus 1. VIII: On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–199.CrossRefGoogle Scholar
Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O., The GL2 main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208.CrossRefGoogle Scholar
Coates, J. and Sujatha, R., On the MH(G)-conjecture, in Non-abelian fundamental groups and Iwasawa theory, London Mathematical Society Lecture Note Series, vol. 393 (Cambridge University Press, Cambridge, 2012), 132–161.Google Scholar
Dokchitser, T. and Dokchitser, V., Computations in non-commutative Iwasawa theory, with an appendix by J. Coates and R. Sujatha, Proc. Lond. Math. Soc. (3) 94 (2006), 211–272.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., On the Birch–Swinnerton-Dyer quotients modulo squares, Ann. of Math. (2) 172 (2010), 567–596.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., Root numbers and parity of ranks of elliptic curves, J. Reine Angew. Math. 658 (2011), 39–64.Google Scholar
Dokchitser, T. and Dokchitser, V., A remark on Tate’s algorithm and Kodaira types, Acta Arith. 160 (2013), 95–100.CrossRefGoogle Scholar
Dokchitser, T. and Dokchitser, V., Local invariants of isogenous elliptic curves, Trans. Amer. Math. Soc. 367 (2015), 4339–4358.CrossRefGoogle Scholar
Drinen, M., Iwasawa 𝜇-invariants of elliptic curves and their symmetric powers, J. Number Theory 102 (2003), 191–213.CrossRefGoogle Scholar
Greenberg, R., Iwasawa theory for elliptic curves, in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Mathematics, vol. 1716 (Springer, Berlin, 1999), 51–144.CrossRefGoogle Scholar
Greenberg, R., Introduction to Iwasawa theory for elliptic curves, in Arithmetic algebraic geometry, IAS/Park City Mathematics Series, vol. 9, eds Conrad, B. et al. (2001), 407–464.Google Scholar
Hachimori, Y. and Matsuno, K., An analogue of Kida’s formula for Selmer groups of elliptic curves, J. Algebraic Geom. 8 (1999), 581–601.Google Scholar
Hachimori, Y. and Venjakob, O., Completely faithful Selmer groups over Kummer extensions, Doc. Math. (2003), 443–478; Extra volume: Kazuya Kato’s fiftieth birthday.Google Scholar
Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adique et applications arithmetiques III, Asterisque, vol. 295 (2004), 117–290.Google Scholar
Kramer, K. and Tunnell, J., Elliptic curves and local 𝜖-factors, Compos. Math. 46 (1982), 307–352.Google Scholar
Kraus, A., Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive, Manuscripta Math. 69 (1990), 353–385.CrossRefGoogle Scholar
Kurihara, M., On the Tate–Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I, Invent. Math. 149 (2002), 195–224.CrossRefGoogle Scholar
Lamplugh, J., An analogue of the Washington–Sinnott theorem for elliptic curves with complex multiplication I, J. Lond. Math. Soc. (2), to appear.Google Scholar
Lockhart, P., Rosen, M. and Silverman, J., An upper bound for the conductor of an abelian variety, J. Algebraic Geom. 2 (1993), 569–601.Google Scholar
Papadopolous, I., Sur la classification de Néron des courbes elliptiques, J. Number Theory 44 (1993), 119–152.Google Scholar
Perrin-Riou, B., Fonctions L p-adique, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), 399–456.CrossRefGoogle Scholar
Ribet, K. A., Torsion points of abelian varieties in cyclotomic extensions, Appendix to N. M. Katz and S. Lang, Finiteness theorems in geometric class field theory, L’Enseign. Math. 27 (1981), 315–319.Google Scholar
Schneider, P., The 𝜇-invariant of isogenies, J. Indian Math. Soc. (N.S.) 52 (1987), 159–170.Google Scholar
Sen, S., Ramification in p-adic Lie extensions, Invent. Math. 17 (1972), 44–50.CrossRefGoogle Scholar
Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
Serre, J.-P., Local fields, Graduate Texts in Mathematics, vol. 67 (Springer, New York, 1979).CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106 (Springer, New York, 1986).CrossRefGoogle Scholar
Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151 (Springer, New York, 1994).CrossRefGoogle Scholar
Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, 18e année, 1965/66, no. 306.Google Scholar
Washington, L., The non-p-part of the class number in a cyclotomic ℤp -extension, Invent. Math. 49 (1978), 87–97.CrossRefGoogle Scholar