Theorie der eisensteinschen reihen von mehreren Veränderlichen

Kloosterman, H. D.

doi:10.1007/bf02940608

Cited by 39 publications

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“…We note that Theorem 1 or 2 naturally extends some known results (for example, Kloostermann [8, Sect.4] and Klingen [7, Satz5]) to the case of weight 1.…”

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confidence: 75%

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A remark on the Hilbert modular forms of weight 1

Shimizu

1983

Math. Ann.

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An elliptic modular form of positive integral weight k is a linear combination of a cusp form and Eisenstein series. This well-known fact was proved by Hecke and was generalized by Kloostermann to the Hilbert modular case (Hecke [5] and Kloostermann [8]). The case k = 1, which was left unsettled in [-8], was solved by Gundlach under certain conditions; a Hilbert modular form for the full modular group over a real quadratic field is a linear combination of a cusp form and Eisenstein series (Gundlach [3,4]). The aim of the present paper is to show that the same holds in general for any principal congruence subgroup over a totally real number field.As one expects by [3], the number of linearly independent Eisenstein series of weight 1 is less than (roughly equal to one half of) the number of cusps. It is enough to prove that the number of linearly independent non-cuspidal forms equals that of linearly independent Eisenstein series. To do this, we shall apply the theory of automorphic representations (the equivalence condition, the uniqueness of Whittaker models, etc .... ) as is developed in Jacquet and Langlands [6].

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“…We note that Theorem 1 or 2 naturally extends some known results (for example, Kloostermann [8, Sect.4] and Klingen [7, Satz5]) to the case of weight 1.…”

mentioning

confidence: 75%

“…In view of [3,8], the Eisenstein series of weight 1 for F(rt; a) should be defined as follows. For (~, 42), (~/~,q2) ~F x F, we say that they are associated modulo n if there exists a u~ a • (n)+ such that qi = u4~ (i= 1,2).…”

Section: Eisenstein Series Of Weightmentioning

confidence: 99%

A remark on the Hilbert modular forms of weight 1

Shimizu

1983

Math. Ann.

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“…(7) obviously remains valid for SfJ. (8) for 5^/ follows from (8), (14). The Fourier expansion of 5^/1 L is obtained from (10) by putting…”

Section: Introductionmentioning

confidence: 99%

“…To define the Eisenstein series (for details see [8]), let p u p 2 eKbe algebraic integers with (p, Pi, p 2 ) = (1), T e H, s e C, and…”

Section: Introductionmentioning

confidence: 99%

On the representation of a number as a sum of squares

Gundlach

1978

Glasgow Math. J.

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It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the formwhere Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact thatis a modular form for a certain congruence subgroup of the modular group, andwithwhere Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

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“…Die P u n k t e ~ : u ~ K unter H i n z u n a h m e von z = nennt m a n Spitzen von F(n). Nach [4] ist die Anzahl der modulo .F(n) in-/iquivalenten Spitzen endlieh, ihre genaue Anzahl betri~gt Dabei ist h die Idealklassenzahl und w(n) die Anzahl der modulo ll inkongruenten Einheiten in K. U m die Fourierentwieklungen einer ganzen Modulform [(~) in der Spitze u zu bekommen, transformiere m a n u naeh ~, Die Matrix A --c d aus K der D e t e r m i n a n t e 1 liiBt sieh dabei so w/ihlen,…”

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�ber den arithmetischen Charakter der Fourierkoeffizienten von Modulformen

Klingen¹

1962

Math. Ann.

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Theorie der eisensteinschen reihen von mehreren Veränderlichen

Cited by 39 publications

References 2 publications

A remark on the Hilbert modular forms of weight 1

A remark on the Hilbert modular forms of weight 1

On the representation of a number as a sum of squares

�ber den arithmetischen Charakter der Fourierkoeffizienten von Modulformen

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