Hyperelliptic modular curves X0(n) and isogenies of elliptic curves over quadratic fields (CROSBI ID 226052)
Prilog u časopisu | izvorni znanstveni rad | međunarodna recenzija
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