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Let FF be a totally real number field FF of degree n>1n>1. Let v1,,vn ⁣:FRv_1,\ldots,v_n \colon F \hookrightarrow \R be the real embeddings of FF, and for aFa \in F abbreviate aivi(a)a_i \colonequals v_i(a) and extend this elementwise to matrices. Write αF>0×\alpha \in F_{>0}^\times if α\alpha is totally positive and ZF,>0×ZF×F>0×\Z_{F,>0}^\times \colonequals \Z_F^\times \cap F_{>0}^\times.

Let k1,,kn2k_1,\dots,k_n \geq 2 be positive integers of the same parity, and let N\mathfrak{N} be a nonzero ideal of the ring of integers ZF\Z_F of FF.

For an ideal bZF\mathfrak{b} \subseteq \Z_F, let γΓ0(N)b{γ=(abcd)GL2(F):cNb, bb1,  and det(γ)ZF,>0×}. \gamma\in\Gamma_0(\mathfrak{N})_\mathfrak{b} \colonequals \left\{\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \GL_2(F) : c \in \mathfrak{Nb}, \ b \in \mathfrak{b}^{-1},\ \text{ and } \det(\gamma) \in \Z_{F,>0}^\times \right\}.

A Hilbert modular form of weight (k1,kn)(k_1\ldots,k_n) and level N\mathfrak{N} is a tuple (fb)b(f_\mathfrak{b})_{\mathfrak{b}} of holomorphic functions f ⁣:HnCf \colon \mathcal{H}^n\rightarrow \C, indexed by ideals b\mathfrak{b} representing the narrow class group of ZF\Z_F, such that for all z=(z1,zn)Hnz=(z_1\ldots,z_n)\in\mathcal{H}^n and all γ=(abcd)Γ0(N)b\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(\mathfrak{N})_\mathfrak{b} we have fb(γz)=fb(a1z1+b1c1z1+d1,,anzn+bncnzn+dn)=i=1n((cizi+di)ki(aidibici)ki/2)fb(z). f_\mathfrak{b}(\gamma z)=f_\mathfrak{b}\left(\frac{a_1z_1+b_1}{c_1z_1+d_1}, \ldots, \frac{a_nz_n+b_n}{c_nz_n+d_n}\right)=\prod_{i=1}^n\left( \frac{(c_iz_i+d_i)^{k_i}}{(a_id_i-b_ic_i)^{k_i/2}}\right)f_\mathfrak{b}(z).

A Hilbert cusp form is a Hilbert modular form that vanishes at the cusps.

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  • Last edited by John Voight on 2024-06-18 08:40:06
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