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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Abhandlungen aus dem...arrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
Article . 2005 . Peer-reviewed
License: Springer TDM
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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 2005
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Congruences for fourier coefficients of lifted siegel modular forms I: Eisenstein lifts

Congruences for Fourier coefficients of lifted Siegel modular forms. I: Eisenstein lifts
Authors: Shin-ichiro Mizumoto;

Congruences for fourier coefficients of lifted siegel modular forms I: Eisenstein lifts

Abstract

The author asks the following question: Does a lifting of modular forms preserve congruences for Fourier coefficients? In this paper he gives an affirmative answer in case of the Eisenstein lift. Let \(S^r_k\) denote the space of cusp forms of even weight \(k> 0\) with respect to the modular group \(\Gamma_r\) of degree \(r\). For \(f\in S^r_k\) and \(Z\) in the upper half space \(\mathbb{H}_n\) of degree \(n\geq r\), the Eisenstein lift \([f]^n_r(Z)\) is defined to be the value at \(s= 0\) of the analytic continuation of \[ [f]^n_r(Z, s)= \sum_M \Biggl({\text{det(Im}(M\langle Z\rangle))\over \text{det(Im}(M\langle Z\rangle^*))}\Biggr)^s\cdot f(M\langle Z\rangle^*)\cdot\text{det}(CZ+ D)^{-k}. \] Here, \(M= \left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)\) runs through a set \(\Delta_{n,r}\setminus\Gamma_n\) of coset representatives as in Klingen's Eisenstein series, and \(M\langle Z\rangle^*\) is the upper left \(r\times r\) block in \(M\langle Z\rangle= (AZ+ B)(CZ+ D)- 1\). The lift can be extended to a linear map \([\cdot]^n_r: M^r_k\to M^n_k\) of the corresponding spaces of all modular forms, and it maps Hecke eigenforms to Hecke eigenforms. Now suppose that \(n> r\), that \(f\in S_k\) and \(g\in S_k\) are Hecke eigenforms with Fourier coefficients in some algebraic number field \(K\), and that \(f\equiv[g]^n_r\pmod{{\mathfrak p}^a}\) for some prime ideal \({\mathfrak p}\) of \(K\) and some \(a> 0\). Then, for certain values \(m\geq n\) and under some assumptions, it follows that \([f]^m_n\equiv [g]^m_r\pmod{{\mathfrak p}^a}\) or \([f]^m_n\equiv 2[g]^m_r\pmod{{\mathfrak p}^a}\). As a consequence one obtains congruences for special values of standard \(L\)-functions. The assumptions are too complicated to be explained here. But they can be verified in some special instances which yield interesting examples, one of which is the following: For \(k\in\{12,16,18,20,22,26\}\), let \(\Delta_k\) be the unique normalized cusp form in \(S^1_k\), and let \(E_k\in M^1_k\) be the normalized Eisenstein series. Then we have \(\Delta_k\equiv-{B_k\over 2k}\cdot E_k\) modulo the numerator of \({B_k\over 2k}\), and it follows that \([\Delta_k]^m_1\equiv -{B_k\over 2k}\cdot E^{(m)}_k\) for \(1\leq m\leq k-2\). There are examples of so-called stable congruences which hold for all \(m\geq n\).

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Japan
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Keywords

Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Congruences for modular and \(p\)-adic modular forms

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
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