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


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 \(2^n\).


Abelian varieties Elliptic curves Ranks 

Mathematics Subject Classification

11G05 11G10 



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.


  1. 1.
    Bosman, J., Bruin, P.J., Dujella, A., Najman, F.: Ranks of elliptic curves with prescribed torsion over number fields. Int. Math. Res. Notices 2014, 2885–2923 (2014)Google Scholar
  2. 2.
    Cohn, P.M.: On the decomposition of a field as a tensor product of fields. Glasg. Math. J. 20, 141–145 (1979)zbMATHCrossRefGoogle Scholar
  3. 3.
    Diem, C., Naumann, N.: On the structure of Weil restrictions of Abelian varieties. J. Ramanujan Math. Soc. 18(2), 153–174 (2003)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Feit, W., Thompson, J.G.: Solvability of groups of odd order. Pac. J. Math. 13, 775–1029 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Frey, G., Jarden, M.: Approximation theory and the rank of Abelian varieties over large algebraic fields. Proc. Lond. Math. Soc. 28, 112–128 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, New York (1993)Google Scholar
  7. 7.
    B.-H. Im and M. Larsen, Infinite rank of elliptic curves over \(\mathbb{Q}^{\rm ab}\). Acta Arith. 158, 49–59 (2013)Google Scholar
  8. 8.
    Mazur, B., Rubin, K., Silverberg, A.: Twisting commutative algebraic groups. J. Algebra 314, 419–438 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Mestre, J.-F.: Rang de courbes elliptiques d’invariant donné. C. R. Acad. Sci. Paris 314, 919–922 (1992)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Mestre, J.-F.: Rang de certaines familles de courbes elliptiques d’invariant donné. C. R. Acad. Sci. Paris 327, 763–764 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Quer, J.: \(\mathbb{Q}\)-curves and abelian varieties of GL\(_2\)-type. Proc. Lond. Math. Soc. 81, 285–317 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Rubin, K., Silverberg, A.: Twists of elliptic curves of rank at least four, in: Ranks of elliptic curves and random matrix theory. London Mathematical Society Lecture Note Series, vol. 341, pp. 177–188. Cambridge University Press, Cambridge (2007)Google Scholar
  13. 13.
    Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–311 (1972)zbMATHMathSciNetCrossRefGoogle Scholar

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

Personalised recommendations