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Are there examples of elliptic curves which has rank 0 over $\mathbb{Q}$, but acquires a high rank ( $\geq 2$) over some quadratic extension?

More generally, are there known bounds for a given extension of degree $n$, how big can the rank be if you start with a rank 0 curve ( over $\mathbb{Q}$ )?

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    $\begingroup$ The first question is easy. Take a curve $A$ over $\mathbb{Q}$ with rank as large as it can get. There exists (plenty of) quadratic twists $E$ of rank $0$. $\endgroup$ Mar 25, 2021 at 19:52
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    $\begingroup$ Based on the previous comment, we should be able to quickly show that the jump in rank could be at least $20$ starting with Elkies - Klagsbrun (2020) Z/2Z curve of rank $20$ posted on Andrej Dujella's page. I'll post the Magma code when I find the suitable quadratic extension. $\endgroup$ Jan 13 at 2:26

2 Answers 2

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I think you will find this paper helpful:

  • Robert J. Lemke Oliver, Frank Thorne, Rank growth of elliptic curves in nonabelian extensions, International Mathematics Research Notices, Volume 2021, Issue 24 (2021) pp 18411–18441, doi:10.1093/imrn/rnz307, arXiv:1810.04018.
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Following the comments, we start with the $\mathbb{Z}/2\mathbb{Z}$ curve $$ E=[ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228, 492395613211229713687666165349208268497505680211462943972719303000171030884176146690352 ] $$ of rank $0$ and extend it over $\mathbb{Q}(\sqrt{2})$ to obtain the $\mathbb{Z}/2\mathbb{Z}$ curve found by Elkies - Klagsbrun (2020) of rank $20$.

The use of Magma's TwoPowerIsogenyDescentRankBound significantly speeds up the process.

SetClassGroupBounds("GRH");
E := EllipticCurve([ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228, 492395613211229713687666165349208268497505680211462943972719303000171030884176146690352 ]);
Coefficients(E);
TwoPowerIsogenyDescentRankBound(E);
QT := MinimalModel(QuadraticTwist(E, 2));
Coefficients(QT);
QT eq EllipticCurve([1,-1,1,-244537673336319601463803487168961769270757573821859853707,961710182053183034546222979258806817743270682028964434238957830989898438151121499931]);

[ 0, 0, 0, -15650411093524454493683423178813553233328484724599030637228,
4923956132112297136876661653492082684975056802114629439727193030001710308841761\
46690352 ]
0 [ 11, 5, 2, 2, 1 ]
[ 7, 5, 2, 2, 1 ]
[ 1, -1, 1, -244537673336319601463803487168961769270757573821859853707,
9617101820531830345462229792588068177432706820289644342389578309898984381511214\
99931 ]
true

A great collection of references on the topic is provided on Andrej Dujella's page High rank elliptic curves with prescribed torsion over quadratic fields. Some of the papers also discuss the extensions over cubic and/or quartic number fields, e.g.

  • J. Bosman, P. Bruin, A. Dujella and F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, Int. Math. Res. Not. 2014 (11) (2014), 2885-2923, doi:10.1093/imrn/rnt013, arXiv:1201.0252, PDF.

  • D. Jeon, C. H. Kim, and Y. Lee, Families of elliptic curves over cubic number fields with prescribed torsion subgroups, Mathematics of Computation, Volume 80, Number 273, January 2011, pp. 579–591, doi:10.1090/S0025-5718-10-02369-0.

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