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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

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1435-5337
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Volume 30, Issue 3

# Period relations for cusp forms of GSp4

Harald Grobner
/ Ronnie Sebastian
Published Online: 2017-08-15 | DOI: https://doi.org/10.1515/forum-2017-0005

## Abstract

Let F be a totally real number field and let π be a cuspidal automorphic representation of ${\mathrm{GSp}}_{4}\left({𝔸}_{F}\right)$, which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period $p\left(\pi \right)$ to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods $p\left(\pi \right)$ and all of their twists $p\left(\pi \otimes \xi \right)$ under arbitrary algebraic Hecke characters ξ. In the appendix, we show that $\left(𝔤,K\right)$-cohomological cusp forms of ${\mathrm{GSp}}_{4}\left({𝔸}_{F}\right)$ all qualify to be of the above type – providing a large source of examples. We expect that these period relations for ${\mathrm{GSp}}_{4}\left({𝔸}_{F}\right)$ will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

MSC 2010: 11F67; 11F41; 11F70; 11F75; 22E55

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Revised: 2017-07-15

Published Online: 2017-08-15

Published in Print: 2018-05-01

The first author is supported by the Austrian Science Fund (FWF), START-prize Y966-N35 and stand-alone-research project P 25974-N25. The second author was partly supported by a DST INSPIRE grant.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 581–598, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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