Let $\Gamma \subset \mathrm{SL}_2(\mathbb{Z})$ be an arithmetic subgroup, and $S_{2k}(\Gamma)$ the space of holomorphic cusp forms of weight $2k$ for $\Gamma$.
Let $\rho_1: \Gamma \rightarrow V_1$ be the standard representation of $\mathrm{GL}_2(\mathbb{R})$ restricted to $\Gamma$, and $\rho_k: \Gamma \rightarrow V_k$ its $k$th symmetric power. Shimura, in the seminal 1959 paper "Sur les intégrales attachées aux formes automorphes", completed the work of Eichler by showing that a certain map $$ S_{2k+2}(\Gamma) \rightarrow H^1_{\mathrm{cusp}}(\mathbb{R}[\Gamma],V_{2k}),\ \ \ f \mapsto [c_f]$$ is an isomorphism of real vector spaces. The codomain is (essentially) group cohomology classes in $H^1(\mathbb{R}[\Gamma],V_{2k})$ that are trivial when restricted to stabilizers of the cusps of $\Gamma$. The map is given by $$ c_f(\gamma) = \int_{z_0}^{\gamma z_0} \mathrm{Re}(\omega_{2k}),$$ where $z_0\in \mathfrak{H}$ is arbitrary, and $$ \omega_n = \left(\begin{array}{c}f(z)dz\\ f(z)zdz\\ \vdots \\ f(z)z^{n}dz\end{array}\right).$$
Since $\Gamma$ is finitely generated and $V_{2k}$ finite dimensional, after choosing generators and a $\Gamma$-stable lattice $L_{2k}\subset V_{2k}$, one may define an integral version $H^1_{\mathrm{cusp}}(\mathbb{Z}[\Gamma],L_{2k})$ of the cohomology group, which turns out to form a full-rank lattice in $H^1_{\mathrm{cusp}}(\mathbb{R}[\Gamma],V_{2k})$. Therefore through the isomorphism, one obtains a lattice $D_{2k}(\Gamma)\subset S_{2k}(\Gamma)$. It is independent of choices up to isogeny, endowing $S_{2k}(\Gamma)$ with a canonical rational structure.
A number of remarkable consequences follow, the most well-known of which is that the eigenvalues of Hecke operators that stabilize $D_{2k}(\Gamma)$ are algebraic integers. The main result however, is the following.
Theorem (Shimura, 1959): The complex torus $$ A_{2k}(\Gamma) = S_{2k}(\Gamma)/D_{2k}(\Gamma)$$ is an abelian variety. Furthermore, there is a ring homomorphism $$ H_k \hookrightarrow \mathrm{End}(A_{2k}(\Gamma)),$$ for a suitable Hecke algebra $H_k$.
The Riemann form on $D_{2k}(\Gamma)$ is obtained by modifying the Peterson inner product to get an alternating, real-valued form, then checking that it takes rational values on $D_{2k}(\Gamma)$.
In the weight two case, as is well-known, $A_2(\Gamma)$ coincides with the Jacobian of the modular curve $X(\Gamma)$, and is therefore defined over $\mathbb{Q}$. Shimura notes that $A_4(\Gamma(3))$ is a CM elliptic curve (hence defined over a number field), then makes the following remark:
Plus généralement, quel que soit le sous-groupe $\Gamma$ de $\mathrm{SL}(2, \mathbb{Z})$, et même s’il n’y a pas de multiplications complexes, on a le droit de penser qu’il existe une relation profonde entre l’arithmétique des séries de Dirichlet attachées aux formes automorphes et nos variétés abéliennes $S_{m}(\Gamma)/D_{m}(\Gamma)$.
Then he moves on to derive certain relations between period integrals of the Ramanujan cusp form, corresponding to the elliptic curve $A_{12}(\Gamma(1))$, which others have studied in more depth and generality since.
My question is as follows:
Are there any general results known about the rationality (or lack thereof) of the abelian varieties $A_{2k}(\Gamma)$, when $k>1$?
If the answer is no, then:
Are there any heuristics or conjectures about when/whether $A_{2k}(\Gamma)$ can be defined over a number field?
If no again,
Are there any general criteria that may be used to effectively check whether $A_{2k}(\Gamma)$ is defined over a number field for some family of non-CM examples?
