A question related to Hilbert modular form

Question

This is a question related to Hilbert modular forms.

Let $\mathbb{K}=\mathbb{Q}(\sqrt D)$ be an imaginary quadratic field with discriminant $D<0$ and $\zeta (\text{mod } m)$ a Hecke character such that $$\zeta((a))= \left( \frac{a}{|a|} \right)^u \text{ if } a \equiv 1 \pmod{m}$$ where $u$ is any non-negative integer. Then $$ f(z)= \sum_a \zeta(a) N_{\mathbb{K}/\mathbb{Q}}(a)^\frac{u}{2} e(zN_{\mathbb{K}/\mathbb{Q}}(a))\in M_k(\Gamma_0(N),\chi) $$ where $k=u+1, N= |D|N_{\mathbb{K}/\mathbb{Q}}(m)$ and $\chi (\text{mod } m)$ is the Dirichlet character given by $$ \chi(n)= \chi_D(n) \text{ if } n\in \mathbb{Z} .$$ Moreover $f$ is a cusp form if $u>0$. This is a theorem from the book "Topics in Classical Automorphic Forms" by Henryk Iwaniec (page 213). This theorem actually provides a connection between the conductor of a Hecke character and the level and weight of a modular form by automorphic induction.

I would like to know if there is a similar thereom in the case of Hilbert modular forms too; that is, if a theorem precisely shows how to construct a hilbert modular form over a totally real field $\mathbb{F}$ from a Hecke character of an imaginary quadratic extension $\mathbb{E}$ of $\mathbb{F}$.

Thank you in advance.

Answer 1

Results of this form are best stated adèlically. Perhaps the canonical reference is this paper of Shalika and Tanaka:

https://doi.org/10.2307/2373316

Sadly the paper was written pre-Jacquet-Langlands and is rather hard to read. Jacquet-Langlands do treat automorphic induction themselves in Section 12 of their seminal book:

http://doi.org/10.1007/BFb0058988

Alternatively, one can try reading this later paper of Labesse and Langlands, which discusses converses to automorphic induction:

https://doi.org/10.4153/CJM-1979-070-3

(See also my answer here: Reference for: CM Hilbert Modular forms arise from Hecke characters)

All of these deal with automorphic induction for Hecke characters; automorphic induction in more general settings is known due to the work of Arthur and Clozel:

https://www.jstor.org/stable/j.ctt1bd6kj6

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In what follows, I summarise the correspondence between Hecke characters and automorphically induced automorphic representations.

Let $E/F$ be a quadratic extension of number fields, and let $\Omega$ be a unitary Hecke character of $\mathbb{A}_E^{\times}$, so that $\Omega$ is the idèlic lift of a classical (primitive) Größencharakter $\psi$ of $E$. This has a completed $L$-function $\Lambda(s,\Omega)$ whose finite part $L(s,\Omega)$ has an Euler product of the form $$\prod_{\mathfrak{P}} \frac{1}{1 - \psi(\mathfrak{P}) \mathrm{N}_{E/\mathbb{Q}}(\mathfrak{P})^{-s}},$$ where the product is over the prime ideals $\mathfrak{P}$ of $\mathcal{O}_E$. Note that $\psi(\mathfrak{P}) = 0$ whenever $\mathfrak{P}$ divides the conductor $\mathfrak{Q}$ of $\Omega$.

Automorphic induction associates to $\Omega$ an automorphic representation $\pi = \pi(\Omega)$ of $\mathrm{GL}_2(\mathbb{A}_F)$ whose completed $L$-function $\Lambda(s,\pi)$ is equal to $\Lambda(s,\Omega)$. (One can prove this via the converse theorem.)

Let $\omega_{\pi}$ denote the central character of $\pi$, so that this is a Hecke character of $\mathbb{A}_F^{\times}$ that is the idèlic lift of a classical (primitive) Größencharakter $\chi_{\pi}$ of $F$; when $F = \mathbb{Q}$, $\chi_{\pi}$ is just a Dirichlet character (it is the nebentypus of the newform associated to $\pi$). One can check that $\omega_{\pi} = \omega_{E/F} \Omega|_{\mathbb{A}_F^{\times}}$, where $\omega_{E/F}$ denotes the quadratic Hecke character associated to the quadratic extension $E/F$. Let $\lambda_{\pi}(\mathfrak{n})$ denote the $\mathfrak{n}$-th Hecke eigenvalue of $\pi$, where $\mathfrak{n}$ is an integral ideal of $\mathcal{O}_F$. (Here I am normalising the Hecke eigenvalues as an analytic number theorist would, namely that $\lambda_{\pi}(\mathfrak{p})$ is the sum of two complex numbers of absolute value $1$ when $\mathfrak{p}$ does not divide the conductor of $\pi$.) Then the finite part $L(s,\pi)$ has an Euler product of the form $$\prod_{\mathfrak{p}} \frac{1}{1 - \lambda_{\pi}(\mathfrak{p}) \mathrm{N}_{F/\mathbb{Q}}(\mathfrak{p})^{-s} + \chi_{\pi}(\mathfrak{p}) \mathrm{N}_{F/\mathbb{Q}}(\mathfrak{p})^{-2s}},$$ where the product is over prime ideals $\mathfrak{p}$ of $\mathcal{O}_F$. Note that the conductor $\mathfrak{q}$ of $\pi$ satisfies $\mathfrak{q} = \mathrm{N}_{E/F}(\mathfrak{Q}) \mathfrak{d}_{E/F}$, where $\mathfrak{d}_{E/F}$ denotes the relative discriminant.

Now for each prime ideal $\mathfrak{p}$, write $\lambda_{\pi}(\mathfrak{p}) = \alpha_{\pi,1}(\mathfrak{p}) + \alpha_{\pi,2}(\mathfrak{p})$, where $\alpha_{\pi,1}(\mathfrak{p}), \alpha_{\pi,2}(\mathfrak{p})$ denote the Satake parameters. Note that $\alpha_{\pi,1}(\mathfrak{p}) \alpha_{\pi,2}(\mathfrak{p}) = \chi_{\pi}(\mathfrak{p})$. Then by comparing Euler products, we have the following:

1. If $\mathfrak{p}$ splits in $E$, so that $\mathfrak{p} \mathcal{O}_E = \mathfrak{P} \sigma(\mathfrak{P})$ for some prime ideal $\mathfrak{P}$ of $\mathcal{O}_E$ with $\mathrm{N}_{E/F}(\mathfrak{P}) = \mathrm{N}_{E/F}(\sigma(\mathfrak{P})) = \mathfrak{p}$, where $\sigma$ denotes the nontrivial Galois automorphism of $E/F$, then $\alpha_{\pi,1}(\mathfrak{p}) = \psi(\mathfrak{P})$ and $\alpha_{\pi,2}(\mathfrak{p}) = \psi(\sigma(\mathfrak{P}))$.
2. If $\mathfrak{p}$ is inert in $E$, so that $\mathfrak{p} \mathcal{O}_E = \mathfrak{P}$ for some prime ideal $\mathfrak{P}$ of $\mathcal{O}_E$ with $\mathrm{N}_{E/F}(\mathfrak{P}) = \mathfrak{p}^2$, then $\alpha_{\pi,1}(\mathfrak{p}) = -\alpha_{\pi,2}(\mathfrak{p}) = \psi(\mathfrak{P})^{1/2}$.
3. If $\mathfrak{p}$ is ramified in $E$, so that $\mathfrak{p} \mid \mathfrak{d}_{E/F}$ and $\mathfrak{p} \mathcal{O}_E = \mathfrak{P}^2$ for some prime ideal $\mathfrak{P}$ of $\mathcal{O}_E$ with $\mathrm{N}_{E/F}(\mathfrak{P}) = \mathfrak{p}$, then $\alpha_{\pi,1}(\mathfrak{p}) = \psi(\mathfrak{P})$ and $\alpha_{\pi,2}(\mathfrak{p}) = 0$.

From this and multiplicativity, one can deduce that $$\lambda_{\pi}(\mathfrak{n}) = \sum_{\substack{\mathfrak{N} \subset \mathcal{O}_E \\ \mathrm{N}_{E/F}(\mathfrak{N}) = \mathfrak{n}}} \psi(\mathfrak{N}).$$

I haven't yet described what happens at the archimedean places. At each archimedean place $w$ of $E$, the local component of $\Omega$ is a unitary character $\Omega_w : E_w^{\times} \to \mathbb{C}^{\times}$ with image in the unit circle.

1. If $E_w \cong \mathbb{R}$, then $\Omega_w(x_w) = \mathrm{sgn}(x_w)^{\kappa_w} |x_w|_w^{it_w}$ for some $\kappa_w \in \{0,1\}$ and $t_w \in \mathbb{R}$. The local component of the completed $L$-function is $\Gamma_{\mathbb{R}}(s + \kappa_w + it_w)$, where $\Gamma_{\mathbb{R}}(s) = \pi^{-s/2} \Gamma(s/2)$.
2. If $E_w \cong \mathbb{C}$, then $\Omega_w(x_w) = e^{i\kappa_w \arg(x_w)} |x_w|_w^{it_w}$ for some $\kappa_w \in \mathbb{Z}$ and $t_w \in \mathbb{R}$. The local component of the completed $L$-function is $\Gamma_{\mathbb{C}}(s + \frac{|\kappa_w|}{2} + it_w)$, where $\Gamma_{\mathbb{C}}(s) = 2(2\pi)^{-s} \Gamma(s)$.

From this, we can describe the local components of $\pi$ at each archimedean place $v$ of $F$.

1. If $F_v \cong \mathbb{R}$ and $v$ splits in $E$ into two real places $w_1$ and $w_2$, then $\pi_v$ is a principal series representation of the form $\mathrm{sgn}^{\kappa_{w_1}} |\cdot|_v^{it_{w_1}} \boxplus \mathrm{sgn}^{\kappa_{w_2}} |\cdot|_v^{it_{w_2}}$.
2. If $F_v \cong \mathbb{R}$ and $v$ ramifies in $E$, so there exists a single complex place lying over $v$, then $\pi_v$ is a discrete series representation of the form $D_{|\kappa_w| + 1} \otimes \left|\det\right|_v^{it_w}$; in particular, the weight is $|\kappa_w| + 1$.
3. If $F_v \cong \mathbb{C}$ then $v$ splits in $E$ into two complex places $w_1$ and $w_2$, and $\pi_v$ is a principal series representation of the form $e^{i\kappa_{w_1} \arg} |\cdot|_v^{it_{w_1}} \boxplus e^{i\kappa_{w_2} \arg} |\cdot|_v^{it_{w_2}}$.

Note that there are restrictions on the parameters $t_w$, since $\Omega$ is trivial on $E^{\times}$ and in particular on $\mathcal{O}_E^{\times}$.

(I write much of this down in section 4 of this paper of mine: https://doi.org/10.1093/imrn/rnx283)

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At this point, you know the Hecke eigenvalues of $\pi$ and also all of its archimedean data. From here, you can write down explicitly the Fourier expansion of the newform of $\pi$ (adèlically, this is its Whittaker expansion). Note that you need to be a little careful, since the constant term in the Fourier expansion does not necessarily vanish: $\pi$ is cuspidal if and only if $\Omega$ does not factor through the norm map; otherwise, the newform associated to $\pi$ is an Eisenstein series.

Answer 2

Belatedly noticed this question... Yes, by this year, as @PeterHumphries said, the fact is an instance (perhaps the simplest) of "automorphic induction". That's the larger picture.

The specific case in the question can be described (and proven) in a much more direct way: binary (pluriharmonic) theta series attached to everywhere locally positive-definite quadratic forms (probably fooling around with generalized idea classes, if done classically). The proof that such things are modular forms is not entirely trivial, but can be (and was) done in a fashion that does not require the full-blown set-up of Segal-Shale-Weil/oscillator repn. (R. Gunning's little orange book on modular forms proves that theta series attached to even-dimensional quadratic forms are elliptic modular forms, for example. Closer examination of the argument leads a person to the theta-correspondence idea... My old book on Hilbert modular forms does a mildly adelic, but not overtly representation-theoretic, proof that Hilbert modular theta series are Hilbert modular forms.)

And, yes, the archimedean theta correspondence sends the trivial repn of real-anisotropic orthogonal groups to holomorphic discrete series. Also, non-trivial repns, attached to spherical harmonics, go to holomorphic discrete series... That's just a local question, answerable by looking for "lowest" K-types.

At unramified finite primes, some computations with Jacquet modules tell which unramified principal series the (local) theta correspondence maps to. (Thinking in terms of the Borel-Casselman-Matsumoto theorem about imbeddings to unramified principal series.)

A big part of the point of "adelizing" is to realize that the same arguments that work over $\mathbb Q$ really do work generally, because most of it is local... As with Iwasawa-Tate for $GL(1)$: once one gets a grip on the slight fanciness, the point is that it's the same shape as Riemann's argument, simply allowing the symbols/terminology to refer to somewhat subtler things. As in $\mathbb A/\mathbb Q$ in place of $\mathbb R/\mathbb Z$.