In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL_2(Z). We ask if we can choose such a parametrisation of an elliptic curve to reflect the field over which it is defined: this can be viewed as a generalisation of the Shimura-Taniyama-Weil conjecture.
Vanishing of L-functions of elliptic curves over number fields
Sums of two biquadrates and elliptic curves of rank $\geq 4$
Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves
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