Rocky Mountain Journal of Mathematics

Transition formulae for ranks of abelian varieties

Daniel Delbourgo and Antonio Lei

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Let $A_{/k}$ denote an abelian variety defined over a number field $k$ with good ordinary reduction at all primes above $p$, and let $K_{\infty }=\bigcup _{n\geq 1} K_n$ be a $p$-adic Lie extension of $k$ containing the cyclotomic $\mathbb{Z}_p$-extension. We use $\mathrm {K}-theory to find recurrence relations for the $\lambda$-invariant at each $\sigma$-component of the Selmer group over $K_{\infty }$, where $\sigma :G_k\rightarrow \mathrm{GL}(V)$. This provides upper bounds on the Mordell-Weil rank for $A(K_n)$ as $n\rightarrow \infty$ whenever $G_{\infty }=\mathrm {Gal}(K_{\infty}/k)$ has dimension at most $3$.

Article information

Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1807-1838.

First available in Project Euclid: 14 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11R23: Iwasawa theory 20F05: Generators, relations, and presentations 22E20: General properties and structure of other Lie groups

Mordell-Weil ranks abelian varieties non-commutative Iwasawa theory representations of pro-$p$ groups $K$-theory


Delbourgo, Daniel; Lei, Antonio. Transition formulae for ranks of abelian varieties. Rocky Mountain J. Math. 45 (2015), no. 6, 1807--1838. doi:10.1216/RMJ-2015-45-6-1807.

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  • John Coates, Takako Fukaya, Kazuya Kato and Ramdorai Sujatha, Root numbers, Selmer groups, and non-commutative Iwasawa theory, J. Alg. Geom. 19 (2010), 19–97.
  • John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob, The $\operatorname{GL}_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Sci. 101 (2005), 163–208.
  • John Coates and Susan Howson, Euler characteristics and elliptic curves, II, J. Math. Soc. Japan 53 (2001), 175–235.
  • John Coates, Peter Schneider and Ramdorai Sujatha, Links between cyclotomic and ${\rm GL}_2$ Iwasawa theory, Doc. Math. (2003), 187–215 (electronic).
  • D. Delbourgo and A. Lei, Estimating the growth in Mordell-Weil ranks and Shafarevich-Tate groups over Lie extensions, Ramanujan J., to appear.
  • John Dixon, Marcus du Sautoy, Avinoam Mann and Dan Segal, Analytic pro-$p$ groups, Second edition, Cambr. Stud. Adv. Math. 61, Cambridge University Press, Cambridge, 1999.
  • Vladimir Dokchitser, Root numbers of non-abelian twists of elliptic curves, Proc. Lond. Math. Soc. 91 (2005), 300–324.
  • Jon González-Sánchez and Benjamin Klopsch, Analytic pro-$p$ groups of small dimensions, J. Group Theory 12 (2009), 711–734.
  • Yoshitaka Hachimori and Otmar Venjakob, Completely faithful Selmer groups over Kummer extensions, Doc. Math. (2003), 443–478 (electronic).
  • Benjamin Klopsch, Pro-$p$ groups with linear subgroup growth, Math. Z. 245 (2003), 335–370.
  • Kazuo Matsuno, Finite $\Lambda$-submodules of Selmer groups of abelian varieties over cyclotomic $\mathbb{Z}_p$-extensions, J. Num. Theor. 99 (2003), 415–443.
  • Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.
  • Jürgen Ritter and Alfred Weiss, Toward equivariant Iwasawa theory. II, Indag. Math. 15 (2004), 549–572.
  • ––––, Toward equivariant Iwasawa theory, III, Math. Ann. 336 (2006), 27–49.
  • Victor Snaith, Explicit Brauer induction, with applications to algebra and number theory, Cambr. Stud. Adv. Math. 40, Cambridge University Press, Cambridge, 1994.