An abelian surface over an imaginary quadratic field has quaternionic multiplication exactly when its residual Galois image is non-split Cartan at two or more primes, under a conjecture that is verified for small levels.
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Non-split Cartan curves and a characterization of fake elliptic curves
An abelian surface over an imaginary quadratic field has quaternionic multiplication exactly when its residual Galois image is non-split Cartan at two or more primes, under a conjecture that is verified for small levels.