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?

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    $\begingroup$ There are two relations. One is the cocycle relation $c(\gamma \tau) = c(\gamma) + \rho_{2k}(\gamma) c(\tau)$, where $\rho_{2k}$ is the representation in the question, which is trivial when $k=0$. The other is the vanishing condition at the cusps. It says for every stabilizer $\gamma$ of every cusp, there exists a vector $v\in V_{2k}$ such that $c(\gamma)=v-\rho_{2k}(v)$. The vector $v$ is allowed to depend on $\gamma$. $\endgroup$ – Zavosh Jan 25 '19 at 5:12
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    $\begingroup$ Right $f(\gamma(z))= (cz+d)^{2k+2} f(z), \gamma'(z) = (cz+d)^{-2}$ so $f(\gamma(z))\gamma(z)^m d\gamma(z)= f(z) (cz+d)^{2k-m} (az+b)^m dz = f(z) (\sum_{l=0}^{2k} \rho(\gamma)_{m,l} z^l) dz $ where $\rho(\gamma)$ is the matrix of your representation $\endgroup$ – reuns Jan 25 '19 at 5:33
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    $\begingroup$ Nice question! I've wondered myself whether Shimura's abelian varieties "mean anything" for $k > 1$ (+ in particular whether they are related to the Kuga--Sato variety inside which you find Galois reps attached to weight 2k mod forms). $\endgroup$ – David Loeffler Jan 25 '19 at 9:12
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    $\begingroup$ Maybe $A_{k}(\Gamma)$ are related to intermediate Jacobians of Kuga-Sato varieties. $\endgroup$ – Zavosh Jan 29 '19 at 23:16

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.

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  • $\begingroup$ I can believe that the complex elliptic curve $E_f$=$\mathbb{C}/L_f$ you can cook up from a weight $2k>2$ cusp eigenform $f$ with $\mathbb{Q}$ coefficients is not defined over $\overline{\mathbb{Q}}$, just because the weight of the motive is wrong. But if $A_k(\Gamma)$ is defined over $\overline{\mathbb{Q}}$, does it mean $E_f$ has to be? $\endgroup$ – Zavosh Jan 29 '19 at 23:29
  • $\begingroup$ In the weight two case of course it's true, but it's usually stated as complex elliptic curves admitting "hyperbolic uniformization of arithmetic type" being defined over $\overline{\mathbb{Q}}$, which seems to rely on the map $X_0(N)\rightarrow J_0(N)$. But maybe that's not necessary. $\endgroup$ – Zavosh Jan 29 '19 at 23:30
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    $\begingroup$ Yes, if an abelian variety $A$ has an isogeny factor that is not defined over $\overline{\mathbb{Q}}$, then it cannot be defined over $\overline{\mathbb{Q}}$: after all, if $A$ is defined over $\overline{\mathbb{Q}}$, the endomorphism algebra of $A$ is defined over $\overline{\mathbb{Q}}$ so the projection onto the elliptic curve is defined over $\overline{\mathbb{Q}}$. (I also think that some of the calculations were or can be done when the space of cusp forms is one-dimensional, whence $A_k(\Gamma)=E_f$.) $\endgroup$ – John Voight Jan 31 '19 at 3:37
  • $\begingroup$ Right, of course. I will accept your answer after a short while, just in case someone else decides to add extra information. $\endgroup$ – Zavosh Jan 31 '19 at 4:15

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