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- W2183529042 abstract "[written Dec 2002] A good source for notation and definitions is the paper [1]. We start by summarising some of this paper. Let F be a totally real field of degree n over Q, with integers O. Note: Shimura uses some other funny letter, not O. Shimura firstly defines “classical” Hilbert modular forms of weight k, as functions on n copies of the upper half plane which transform well under elements of a congruence subgroup Γ of GL2 (F ), the invertible matrices with totally positive determinant (see below for his definition of a congruence subgroup). His definition involves the k/2th power of a determinant; this is OK because his determinant is always totally positive and he takes the positive roots. In particular his centre always acts trivially. Let Mk(Γ) denote the space of such forms. The definition of congruence subgroup that Shimura uses is: Γ ⊂ GL2 (F ) is a congruence subgroup if it contains all the matrices in SL2(O) congruent to the identity mod N for some ideal N of O, and if furthermore the image of Γ in PGL2(F ) is commensurable with the image of SL2(F ). So there is a finite index subgroup of the totally positive units in OF such that Γ contains diag(u, u−1) for all u in this subgroup. On the other hand we don’t know much at all about the units u such that diag(u, u) ∈ Γ; this could hold just for u = 1 or alternatively it could hold for any unit; both SL2(O) and GL2 (O) are congruence subgroups. This is OK for him—the associated spaces of modular forms are the same—because his centre acts trivially. Such Hilbert modular forms have a Fourier expansion ∑ ξ c(ξ)eF (ξz) where ξ ranges over 0 and the totally positive elements of a lattice in F , c(ξ) are complex numbers, and eF (z) is e 2πi ∑ ν zν . It is formal to check that c(uξ) = uc(ξ) if diag(u, u−1) ∈ Γ and moreover that c(uξ) = uc(ξ) if diag(u, 1) ∈ Γ (note that this implies that u is totally positive). Taking direct limits over all congruence subgroups Γ and get Mk. Remark: this space is zero unless either all the ki are non-negative and the same, or they are all positive. On the other hand there’s no reason why they should" @default.
- W2183529042 created "2016-06-24" @default.
- W2183529042 creator A5020985134 @default.
- W2183529042 date "2012-01-01" @default.
- W2183529042 modified "2023-09-27" @default.
- W2183529042 title "An example of a non-paritious Hilbert modular form." @default.
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- W2183529042 hasPublicationYear "2012" @default.
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