STNB2026 (39th edition)
Recognizing fake elliptic curves
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Abstract
The system of Galois representations attached to an abelian surface with quaternionic multiplication looks remarkably similar to the one attached to an elliptic curve. The two systems are two-dimensional and have rational traces, and they both satisfy an open-image theorem. For this reason, QM surfaces are sometimes called fake elliptic curves. In this talk, I will explain a result over imaginary quadratic fields that separates fake elliptic curves from actual ones by studying the non-surjective primes of the residual representations.
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