Rocky Mountain Journal of Mathematics

Transition formulae for ranks of abelian varieties

Daniel Delbourgo and Antonio Lei

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Abstract

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

Subjects
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

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

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


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