Glasnik Matematicki, Vol. 50, No. 2 (2015), 333-348.

ON THE NUMBER OF N-ISOGENIES OF ELLIPTIC CURVES OVER NUMBER FIELDS

Miljen Mikić and Filip Najman

Kumičićeva 20, 51000 Rijeka, Croatia
e-mail: miljen.mikic@gmail.com

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: fnajman@math.hr

Abstract.   We find the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X0(n). We show that for n=14,15,20,21,49 there exist infinitely many quartic fields K such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞ . In the case n=27 we prove that there are infinitely many sextic fields such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞.

2010 Mathematics Subject Classification.   11G05, 11G18, 11R16, 14H52.

Key words and phrases.   Elliptic curves, Mordell-Weil group, isogenies.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.05

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