Naslov
Hyperelliptic modular curves X0(n) and isogenies of elliptic curves over quadratic fields

Autori
Bruin, Peter ; Najman, Filip

Izvornik
LMS Journal of Computation and Mathematics (1461-1570) 18 (2015), 1; 578-602

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
elliptic curve ; isogeny ; quadratic field

Sažetak
We study elliptic curves over quadratic fields with isogenies of certain degrees. Let n be a positive integer such that the modular curve X0(n) is hyperelliptic of genus ⩾2 and such that its Jacobian has rank 0 over Q. We determine all points of X0(n) defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to Q¯¯¯- isomorphism, every elliptic curve over a quadratic field K admitting an n-isogeny is d- isogenous, for some d∣n, to the twist of its Galois conjugate by a quadratic extension L of K. We determine d and L explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to Q¯¯¯- isomorphism, all elliptic curves with n- isogenies over quadratic fields are in fact Q- curves.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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