# Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

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## Abstract

Let \(E_{/\mathbb {Q}}\!\) be an elliptic curve, \(p>3\) a good ordinary prime for *E*, and \(K_\infty \) a *p*-adic Lie extension of a number field *k*. Under some standard hypotheses, we study the asymptotic growth in both the Mordell–Weil rank and Shafarevich–Tate group for *E* over a tower of extensions \(K_n/k\) inside \(K_\infty \); we obtain lower bounds on the former, and upper bounds on the latter’s size.

### Keywords

Elliptic curves Mordell–Weil ranks Noncommutative Iwasawa theory### Mathematics Subject Classification

Primary 11R23 11G05 22E20 22E05## Notes

### Acknowledgments

The majority of this work was carried out during the first named author’s visit to Université Laval in May–June 2015, and he would like to thank them for their generous hospitality, and in particular Hugo Chapdelaine. The authors would also like to thank Wei Lu for his comments on an earlier version of this article.

### References

- 1.Darmon, H.: Tian, Ye: Heegner points over towers of Kummer extensions. Can. J. Math.
**62**(5), 1060–1081 (2010)CrossRefMATHGoogle Scholar - 2.Dokchitser, T.: Dokchitser, Vladimir: Regulator constants and the parity conjecture. Invent. Math.
**178**(1), 23–71 (2009)MathSciNetCrossRefMATHGoogle Scholar - 3.Greenberg, R.: Iwasawa Theory, Projective Modules, and Modular Representations, vol. 211, no. 992. Memoirs of the American Mathematical Society (2011)Google Scholar
- 4.Guo, L.: General Selmer groups and critical values of Hecke \(L\)-functions. Math. Ann.
**297**(2), 221–233 (1993)MathSciNetCrossRefMATHGoogle Scholar - 5.Greenberg, R.: Galois theory for the Selmer group of an abelian variety. Compos. Math.
**136**(3), 255–297 (2003)MathSciNetCrossRefMATHGoogle Scholar - 6.Cuoco, A.A., Monsky, P.: Class numbers in \({\mathbf{Z}}^{d}_{p}\)-extensions. Math. Ann.
**255**(2), 235–258 (1981)MathSciNetCrossRefMATHGoogle Scholar - 7.Perbet, G.: Sur les invariants d’Iwasawa dans les extensions de Lie \(p\)-adiques. Algebr. Number Theory
**5**(6), 819–848 (2011)MathSciNetCrossRefMATHGoogle Scholar - 8.Gonzalez-Sanchez, J., Klopsch, B.: Analytic pro-p groups of small dimensions. J. Group Theory
**12**(5), 711–734 (2009)MathSciNetCrossRefMATHGoogle Scholar - 9.Delbourgo, D., Lei, A.: Transition formulae for ranks of abelian varieties. Rocky Mt. J. Math.
**45**(6), 1807–1838 (2015)MathSciNetCrossRefMATHGoogle Scholar - 10.Coates, J., Fukaya, T., Kato, K., Sujatha, R.: Root numbers, Selmer groups, and non-commutative Iwasawa theory. J. Algebr. Geom.
**19**(1), 19–97 (2010)MathSciNetCrossRefMATHGoogle Scholar - 11.Dokchitser, V.: Root numbers of non-abelian twists of elliptic curves. Proc. Lond. Math. Soc. (3)
**91**, 300–324 (2005). With an appendix by Tom FisherMathSciNetCrossRefMATHGoogle Scholar - 12.Delbourgo, D.: Peters, Lloyd: Higher order congruences amongst Hasse-Weil \(L\)-values. J. Aust. Math. Soc.
**98**(1), 1–38 (2015)MathSciNetCrossRefMATHGoogle Scholar - 13.Zhang, S.W.: Gross-Zagier Formula for \({{\rm GL}}(2)\). II, Heegner Points and Rankin \(L\)-Series. Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge (2004)Google Scholar
- 14.Skinner, C., Urban, E.: The Iwasawa main conjectures for \({{\rm GL}}_2\). Invent. Math.
**195**(1), 1–277 (2014)MathSciNetCrossRefMATHGoogle Scholar - 15.Zerbes, S.L.: Selmer groups over \(p\)-adic Lie extensions. J. Lond. Math. Soc.
**70**(3), 586–608 (2004)MathSciNetCrossRefMATHGoogle Scholar - 16.Harris, M.: \(p\)-Adic representations arising from descent on abelian varieties. Compos. Math.
**39**(2), 177–245 (1979)MathSciNetMATHGoogle Scholar - 17.Coates, J.: Fragments of the \({{\rm GL}}_2\) Iwasawa theory of elliptic curves without complex multiplication. In: Arithmetic Theory of Elliptic Curves (Cetraro, 1997). Lecture Notes in Mathematics, vol. 1716, pp. 1–50. Springer, Berlin (1999)Google Scholar
- 18.Serre, J.P.: Cohomologie Galoisienne, Cours au Collège de France, vol. 1962. Springer, Berlin (1962/1963)Google Scholar
- 19.Hachimori, Y., Venjakob, O.: Completely faithful Selmer groups over Kummer extensions. Kazuya Kato’s fiftieth birthday. Doc. Math., Extra vol., 443–478 (2003) (electronic)Google Scholar
- 20.Burns, D., Venjakob, O.: On descent theory and main conjectures in non-commutative Iwasawa theory. J. Inst. Math. Jussieu
**10**(1), 59–118 (2011)MathSciNetCrossRefMATHGoogle Scholar - 21.Howson, S.: Euler characteristics as invariants of Iwasawa modules. Proc. Lond. Math. Soc.
**3**(85), 634–658 (2002)MathSciNetCrossRefMATHGoogle Scholar - 22.Venjakob, O.: On the structure theory of the Iwasawa algebra of a \(p\)-adic Lie group. J. Eur. Math. Soc. (JEMS)
**4**(3), 271–311 (2002)MathSciNetCrossRefMATHGoogle Scholar - 23.Ritter, J., Weiss, A.: Toward equivariant Iwasawa theory. III. Math. Ann.
**336**(1), 27–49 (2006)MathSciNetCrossRefMATHGoogle Scholar - 24.Bourbaki, N.: Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs, Actualités Scientifiques et Industrielles, vol. 1314. Hermann, Paris (1965)Google Scholar
- 25.Kobayashi, S.: Iwasawa theory for elliptic curves at supersingular primes. Invent. Math.
**152**(1), 1–36 (2003)MathSciNetCrossRefMATHGoogle Scholar - 26.Coates, J.: Elliptic Curves—The Crossroads of Theory and Computation. Lecture Notes in Computer Science, vol. 2369, pp. 9–19 (2002)Google Scholar
- 27.Matsuno, K.: Finite \(\Lambda \)-submodules of Selmer groups of abelian varieties over cyclotomic \({\mathbb{Z}}_p\)-extensions. J. Number Theory
**99**(2), 415–443 (2003)MathSciNetCrossRefMATHGoogle Scholar - 28.Coates, J., Schneider, P., Sujatha, R.: Links between cyclotomic and \({\text{ GL }}_2\) Iwasawa theory. Kazuya Kato’s fiftieth birthday. Doc. Math. Extra vol., 187–215 (2003) (electronic)Google Scholar
- 29.Delbourgo, D., Lei, A.: Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction. Math. Proc. Cambridge Philos. Soc.
**160**(1), 11–38 (2016)MathSciNetCrossRefGoogle Scholar - 30.Venjakob, O.: A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math.
**559**, 153–191 (2003). With an appendix by Denis VogelMathSciNetMATHGoogle Scholar