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The Ramanujan Journal

, Volume 39, Issue 2, pp 259–269 | Cite as

The growth of the rank of Abelian varieties upon extensions

  • Pieter Bruin
  • Filip Najman
Article

Abstract

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties with respect to finite extensions of number fields. First, we show that if \(A\) is an Abelian variety over a number field \(K\) and \(L/K\) is a finite Galois extension such that \({{\mathrm{Gal}}}(L/K)\) does not have an index 2 subgroup, then \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) can never be 1. We show that \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) is either 0 or \(\ge p-1\), where \(p\) is the smallest prime divisor of \(\# {{\mathrm{Gal}}}(L/K)\), and we obtain more precise results when \({{\mathrm{Gal}}}(L/K)\) is alternating, \({{\mathrm{SL}}}_2(\mathbb {F}_p)\) or \({{\mathrm{PSL}}}_2(\mathbb {F}_p)\) for \(p>2\). This implies a restriction on \({{\mathrm{rk}}}E(K(E[p]))-{{\mathrm{rk}}}E(K(\zeta _p))\) when \(E/K\) is an elliptic curve whose mod \(p\) Galois representation is surjective. We obtain similar results for the growth of the rank over certain non-Galois extensions. Second, we show that for every \(n\ge 2\) there exists an elliptic curve \(E_n\) over a number field \(K_n\) such that \(\mathbb {Q}\otimes {{\mathrm{End}}}_\mathbb {Q}{{\mathrm{Res}}}_{K_n/\mathbb {Q}} E_n\) contains a number field of degree \(2^n\). We ask whether every elliptic curve \(E/K\) has infinite rank over \(K\mathbb {Q}(2)\), where \(\mathbb {Q}(2)\) is the compositum of all quadratic extensions of \(\mathbb {Q}\). We show that if the answer is yes, then for any \(n\ge 2\), there exists an elliptic curve \(E_n\) over a number field \(K_n\) admitting infinitely many quadratic twists whose rank is a positive multiple of \(2^n\).

Keywords

Abelian varieties Elliptic curves Ranks 

Mathematics Subject Classification

11G05 11G10 

Notes

Acknowledgments

We are greatly indebted to Jordi Quer for help with the proof of Proposition 9. We are grateful to Kęstutis Česnavičius, Andrej Dujella, Ivica Gusić, Matija Kazalicki and Michael Larsen for many helpful comments.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Mathematisch InstituutUniversiteit LeidenLeidenNetherlands
  3. 3.Department of MathematicsUniversity of ZagrebZagrebCroatia

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