Abstract
A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every ℚ-curve is modular, and he showed that a positive answer would follow from J.-P. Serre's conjecture on mod p Galois representations. We answer Ribet's question in the affirmative, subject to certain local conditions at 3.
Citation
Jordan S. Ellenberg. Chris Skinner. "On the modularity of ℚ-curves." Duke Math. J. 109 (1) 97 - 122, 15 July 2001. https://doi.org/10.1215/S0012-7094-01-10914-9
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