Poincar�sche und Eisensteinsche Reihen zur Hilbertschen Modulgruppe

Gundlach, Karl-Bernhard

doi:10.1007/bf01166576

Cited by 14 publications

(11 citation statements)

References 4 publications

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“…A proof in the case n = (1) follows immediately from [3] and (2.5). Namely, let G~ b denote the Eisenstein series (3.1) with r =~2=0, n= (1).…”

Section: Theorem 1 At(f(n;a)) Is Spanned By the Eisenstein Series (3mentioning

confidence: 94%

“…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%

“…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.…”

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confidence: 95%

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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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“…A proof in the case n = (1) follows immediately from [3] and (2.5). Namely, let G~ b denote the Eisenstein series (3.1) with r =~2=0, n= (1).…”

Section: Theorem 1 At(f(n;a)) Is Spanned By the Eisenstein Series (3mentioning

confidence: 94%

Section: Eisenstein Series Of Weightmentioning

confidence: 99%

mentioning

confidence: 95%

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A remark on the Hilbert modular forms of weight 1

Shimizu

1983

Math. Ann.

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“…The last three pairs have p u p 2 independent mod(2) over 1; in these cases, according to Theorem 5, the functions y i G r (z; (2),p lt p 2 ;K) for r > l are cusp forms from A^2r(r Q (2), 1). From Lemma 1 and Theorem 5 we deduce that the functions g T (T;K) = G r (r;(2),l,e 0 ;K), …”

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confidence: 97%

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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“…Proceeding from [H] where the same groups naturally arise, Gundlach [Gl,G2] investigated Hubert modular forms for the groups rlb/. In fact, he developed a rather full theory of Eisenstein series and Poincare series, including estimates on the dimension of the space of cusp forms of given weight.…”

Section: )mentioning

confidence: 99%

𝑝-adic hyperbolic planes and modular forms

Rhodes¹

1994

Trans. Amer. Math. Soc.

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Abstract. For K a number field and p a finite prime of K, we define a p-adic hyperbolic plane and study its geometry under the action of GL2(Kr). Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator Tv . Letting 5 be a finite set of primes of a totally real K (including all the infinite ones), a modular group T(S) is defined. This group acts discontinuously on a product of classical and p-adic hyperbolic planes. S-modular forms and their associated Dirichlet series are studied.

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Poincar�sche und Eisensteinsche Reihen zur Hilbertschen Modulgruppe

Cited by 14 publications

References 4 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

𝑝-adic hyperbolic planes and modular forms

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