# Potential automorphy of abelian varieties

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?

• Could you tell what results you assume about $\rho : \text{Gal}(\overline{K}/K) \to \text{Aut}(T_{\ell}(A))$ ? Nov 20, 2017 at 4:20
• Why do you think the Artin conjecture is relevant? Nov 20, 2017 at 9:56
• @reuns: No assumptions on $\rho$. Nov 20, 2017 at 18:49
• No assumptions but plenty of results have been proven, so please remind us (for those who know Artin L-functions better than abelian varieties) Nov 20, 2017 at 18:59
• @WillSawin: Because the L-function of $A$ over any Galois extension $K'$ of $K$ will be a product of the L-function $L(A \otimes \chi, s)$ where $\chi$ is an irreducible representation of the finite Galois group $\mathrm{Gal}(K'/K)$. So it seemed the Artin conjecture for $\chi$ might be relevant here. Nov 21, 2017 at 2:19

(ii) has nothing to do with Artin's conjecture I believe. It is known that (1) implies (2) in the case $K'/K$ solvable. This is the so-called solvable base change, solved by Arthur and Clozel in the 1980's, and it really has nothing to do with abelian varieties. The general non-solvable case is wide-open.
If you know that $A$ is automorphic and $\chi$ is automorphic then from the Rankin-Selberg (Jacquet, Piatetski-Shapiro, Shalika) you know that the L-function $L(A \times \chi,s)$ is meromorphic and satisfies the expected functional equation. That gives a weak form of (ii). (I don't know the answer to your other questions.)
• Do you mean "the L-function associated to $A$ (resp. $\chi$ )" instead of $A$ and $\chi$ themselves ? Nov 23, 2017 at 19:12