Non-split Cartan curves and a characterization of fake elliptic curves

Abstract

The $\ell$-adic Tate module of an abelian surface with quaternionic multiplication (QM) decomposes as two copies of a two-dimensional $\mathbb{Q}$-rational representation. To this day, there is no criterion to distinguish these representations from those attached to elliptic curves. For this reason, QM abelian surfaces are usually called fake elliptic curves. In this paper we characterize QM abelian surfaces defined over an imaginary quadratic field $K$. Namely, we consider an abelian surface $A/K$ without potential CM and whose $L$-function is a square. Under a reasonable conjecture, we show that $A$ has QM if and only if it has residual image contained in a non-split Cartan group modulo at least two primes. The characterization is unconditional for all indefinite quaternion discriminants up to 33. The proof is based on previous work of Siksek and Michaud-Jacobs on quadratic points on non-split Cartan modular curves. In particular, we prove that all quadratic points on $X_{ns}(6)$, $X_{ns}(10)$ and $X_{ns}(15)$ are non-exceptional.