Let $ A$ be an abelian variety over $ \mathbb Q$ . One could ask

(1) is there a finite extension $ K$ of $ \mathbb Q$ such that the L-function $ L(A/K,s)$ is the L-function of an automorphic form?

or

(2) is there a finite extension $ K$ of $ \mathbb Q$ such that, for every finite extension $ K \subset K’$ , the L-function $ L(A/{K’},s)$ of $ A$ over $ K’$ is the L-function of an automorphic form?

Questions: (i) It seems potential automorphic refers to (1). Is that correct?

(ii) Does (1) imply (2) under the assumption of the Artin conjecture?

For elliptic curves over $ \mathbb Q$ , modularity is equivalent to a non-constant map from the modular curve to the given elliptic curve and hence a cycle on the product of the modular curve and the elliptic curve.

(iii) If an abelian variety is automorphic, then is an appropriate algebraic cycle expected on the product of a Shimura variety and the given abelian variety?