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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.

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  • 2
    $\begingroup$ Yes. This is automorphic induction. $\endgroup$ – Peter Humphries Jul 20 '20 at 23:07
  • $\begingroup$ Please provide some references. $\endgroup$ – Kiddo Jul 21 '20 at 21:36
  • $\begingroup$ Did you try googling automorphic induction? For example, see section 13 of doi.org/10.2307/2944321 $\endgroup$ – Peter Humphries Jul 21 '20 at 22:33
  • $\begingroup$ If you are unsatisfied with my answer, please do let me know what you are instead looking for. (Also, you should accept answers to your questions when they are satisfactory - for some reason, you have only done this once out of the ten questions that you have asked.) $\endgroup$ – Peter Humphries Jul 30 '20 at 0:26
  • $\begingroup$ Hi @PeterHumphries ,thanks a lot for the answer. Last few weeks I was busy with some other works and couldn't manage to give it enough time. I will surely accept the answer once I completely understand it. $\endgroup$ – Kiddo Aug 20 '20 at 21:59
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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


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)


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.

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