�ber den arithmetischen Charakter der Fourierkoeffizienten von Modulformen

Klingen, Helmut

doi:10.1007/bf01470949

Cited by 24 publications

(12 citation statements)

References 7 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.…”

mentioning

confidence: 75%

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%

A remark on the Hilbert modular forms of weight 1

Shimizu

1983

Math. Ann.

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“…Although often only implicit, this mechanism is an important recurring theme in the theory of automorphic forms, with both analytic and arithmetic applications: from among the enormous range of range of literature from the early [30,27], one may mention [17,18,31,32] (and many other later papers of Shimura), also [4,5], and [9,7], and the partial survey of examples of the Rankin-Selberg method in [3]. The most typical scenario involves part of a spectral decomposition of the restriction of an Eisenstein series to a smaller reductive group.…”

Section: Applicationsmentioning

confidence: 99%

Hecke duality of Ikeda lifts

Garrett

Heim

2015

Journal of Number Theory

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“…A classical theorem of Siegel [40], Klingen [31] and Shintani [39] states that Θ S := Θ S (0) lies in Q[G]. This was refined by Deligne-Ribet [23] and Cassou-Noguès [4], who proved that under a certain mild technical condition on T discussed in (19) below, we have Θ S,T := Θ S,T (0) ∈ Z [G].…”

Section: The Brumer-stark Conjecturementioning

confidence: 99%

“…Denote the image of the vectors b 0 and b 1 in ∇Σ ′ Σ (L) by b 0 and b 1 , respectively. Recall the R-algebra R L defined in (31). We define…”

Section: The Module ∇ Lmentioning

confidence: 99%

Brumer-Stark Units and Hilbert's 12th Problem

Dasgupta¹,

Kakde²

2021

Preprint

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Let F be a totally real field of degree n and p an odd prime. We prove the ppart of the integral Gross-Stark conjecture for the Brumer-Stark p-units living in CM abelian extensions of F . In previous work, the first author showed that such a result implies an exact p-adic analytic formula for these Brumer-Stark units up to a bounded root of unity error, including a "real multiplication" analogue of Shimura's celebrated reciprocity law in the theory of Complex Multiplication. In this paper we show that the Brumer-Stark units, along with n − 1 other easily described elements (these are simply square roots of certain elements of F ) generate the maximal abelian extension of F . We therefore obtain an unconditional solution to Hilbert's 12th problem for totally real fields, albeit one that involves p-adic integration, for infinitely many primes p.Our method of proof of the integral Gross-Stark conjecture is a generalization of our previous work on the Brumer-Stark conjecture. We apply Ribet's method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module ∇ L that incorporates an integral version of the Greenberg-Stevens L -invariant into the theory of Ritter-Weiss modules. This allows for the reinterpretation of Gross's conjecture as the vanishing of the Fitting ideal of ∇ L . This vanishing is obtained by constructing a quotient of ∇ L whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms.

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�ber den arithmetischen Charakter der Fourierkoeffizienten von Modulformen

Cited by 24 publications

References 7 publications

A remark on the Hilbert modular forms of weight 1

A remark on the Hilbert modular forms of weight 1

Hecke duality of Ikeda lifts

Brumer-Stark Units and Hilbert's 12th Problem

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