Elliptic ℚ-curves: restriction of scalars and embedding problems

Jordi Quer

January 2001

Elliptic ℚ-curves are elliptic curves defined over number fields that are isogenous to all its Galois conjugates. It is conjectured that ℚ-curves are precisely the elliptic curves that are quotients of modular curves X_{1}(N), and several results have been obtained in that direction in the last years (Hasegawa-Hashimoto-Momose, Hida, Ellenberg-Skinner), using the ideas and results of Wiles and others for proving the modularity of elliptic curves over ℚ.

All the non-CM elliptic ℚ-curves can be obtained, up to isogeny, from the rational points on the modular curves X^{∗}(N) quotient of X_{0}(N) by the group of Atkin-Lehner involutions, for squarefree values of N. In this way one obtains the j-invariants of ℚ-curves up to isogeny.

In the isogeny class of every elliptic ℚ-curve there are representatives having the property that, by restriction of scalars, one obtains an abelian variety isogenous to a product of varieties of GL_{2}-type. In case that the curve is modular, this means that the L-series of the elliptic curve over the number field is a product of L-series of newforms for Γ_{1}(N).

In practice, if one starts from the j-invariants obtained parametrizing the rational points of X^{∗}(N), the computation of isogenous curves having the property that the restriction of scalars is isogenous to a product of abelian varieties of GL_{2}-type, requires to solve a central embedding problem in Galois theory corresponding to embed an abelian extension of ℚ into an extension of double degree.

In this talk we first discuss some general facts on ℚ-curves concerning their parametrization up to isogeny, their fields of definition, and properties of the abelian varieties obtained by restriction of scalars. Then, we give a way to solve the kind of embedding problems described in the last paragraph, and as an application we find several examples of ℚ-curves with restriction of scalars a product of abelian varieties of GL_{2}-type.