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?** 



Shimura's construction of an abelian variety from cusp forms of weight $2k$
Hmm, see Remark 2.3.2 on page 15 in the thesis of Kimberly Hopkins (<https://repositories.lib.utexas.edu/bitstream/handle/2152/ETD-UT-2010-05-1423/HOPKINS-DISSERTATION.pdf>): she computes some elliptic curve factors of these quotients and her computations suggest that the j-invariants are transcendental (as expected). I don't know how to check that a complex number computed to some precision is a transcendental number, so I am pessimistic about an answer to your final question. 


