# A question related to Hilbert modular form

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 $$\mathbb{F}$$.

• Yes. This is automorphic induction. Jul 20, 2020 at 23:07
• Please provide some references. Jul 21, 2020 at 21:36
• Did you try googling automorphic induction? For example, see section 13 of doi.org/10.2307/2944321 Jul 21, 2020 at 22:33
• 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.) Jul 30, 2020 at 0:26
• 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. Aug 20, 2020 at 21:59

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

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.