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Hyperelliptic modular curves X0(n) and isogenies of elliptic curves over quadratic fields (CROSBI ID 226052)

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Bruin, Peter ; Najman, Filip Hyperelliptic modular curves X0(n) and isogenies of elliptic curves over quadratic fields // LMS Journal of Computation and Mathematics, 18 (2015), 1; 578-602. doi: 10.1112/S1461157015000157

Podaci o odgovornosti

Bruin, Peter ; Najman, Filip

engleski

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

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.

elliptic curve ; isogeny ; quadratic field

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Podaci o izdanju

18 (1)

2015.

578-602

objavljeno

1461-1570

10.1112/S1461157015000157

Povezanost rada

Matematika

Poveznice
Indeksiranost