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?

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