# Shimura's construction of an abelian variety from cusp forms of weight $2k$

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?

• 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 $\gamma$. Jan 25, 2019 at 5:12
• 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 Jan 25, 2019 at 5:33
• 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). Jan 25, 2019 at 9:12
• Maybe $A_{k}(\Gamma)$ are related to intermediate Jacobians of Kuga-Sato varieties. Jan 29, 2019 at 23:16

• 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? Jan 29, 2019 at 23:29
• 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. Jan 29, 2019 at 23:30
• 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 $A_k(\Gamma)=E_f$.) Jan 31, 2019 at 3:37