14
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 3
    $\begingroup$ 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
  • 1
    $\begingroup$ Maybe $A_{k}(\Gamma)$ are related to intermediate Jacobians of Kuga-Sato varieties. $\endgroup$ – Zavosh Jan 29 '19 at 23:16
6
+200
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.