Parametrization of virtually $K$-rational Drinfeld modules of rank two

Abstract

For an algebraic extension K of the rational function field Fq(T ) over a finite field, we introduce the notion of K-virtual Drinfeld modules as a function field analogue of Q-curves, which are elliptic curves over Q isogenous to all its Galois conjugates. Our goal in this article is to prove that all K-virtual Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by K-rational points of a quotient curve of the Drinfeld modular curve Y0(n) with some square-free level n, which is an analogue of Elkies' well-known result on Q-curves.