Hyperelliptic modular curves and isogenies of elliptic curves over quadratic fields

HYPERELLIPTIC MODULAR CURVES X0(n) AND ISOGENIES OF ELLIPTIC CURVES OVER QUADRATIC FIELDS

Let n be an integer such that the modular curve X0(n) is hyperelliptic of genus ≥ 2 and such that the Jacobian of X0(n) 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. As a consequence, we show that up to Q-isomorphism, all but finitely many elliptic curves with n-isogenies over quadratic fields are in fact ...

Elliptic and hyperelliptic curves over supersimple fields

We prove that if F is an infinite field with characteristic different from 2, whose theory is supersimple, and C is an elliptic or hyperelliptic curve over F with generic moduli then C has a generic F -rational point. The notion of generity here is in the sense of the supersimple field F .

Modular Forms and Elliptic Curves over Imaginary Quadratic Fields

On Hyperelliptic Modular Curves over Function Fields

We prove that there are only finitely many modular curves of Delliptic sheaves over Fq(T ) which are hyperelliptic. In odd characteristic we give a complete classification of such curves.

An Algorithm for Modular Elliptic Curves over Real Quadratic Fields

Let F be a real quadratic field with narrow class number one, and f a Hilbert newform of weight 2 and level n with rational Fourier coefficients, where n is an integral ideal of F . By the Eichler-Shimura construction, which is still a conjecture in many cases when [F : Q] > 1, there exists an elliptic curve Ef over F attached to f . In this paper, we develop an algorithm that computes the (can...