In describing, say, the moduli of Shimura curves, people often refer to "fake" or "false elliptic curves" ("les fausses courbes elliptiques"), which are abelian surfaces whose endomorphism ring is an order of quaternion algebra with center $\textbf{Q}$.
Where does this terminology come from? (Deligne-Rapoport attributes this terminology to Serre, but I cannot seem to find a paper that gives an explanation for the term.) And why should one distinguish this specific case as a "false elliptic curve" and not, say, some other abelian variety with some special kind of endomorphism ring?