Rocky Mountain Journal of Mathematics

Transition formulae for ranks of abelian varieties

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 Source Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1807-1838. Dates First available in Project Euclid: 14 March 2016 Permanent link to this document https://projecteuclid.org/euclid.rmjm/1457960336 Digital Object Identifier doi:10.1216/RMJ-2015-45-6-1807 Mathematical Reviews number (MathSciNet) MR3473156 Zentralblatt MATH identifier 1355.11070 Citation 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. https://projecteuclid.org/euclid.rmjm/1457960336 References • 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.
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