---
_id: '53191'
abstract:
- lang: eng
text: "This paper is the first in a series of two dedicated to the study of period
relations of the type \r\n\r\n \r\n \r\n
\ L\r\n \r\n \r\n (\r\n
\ \r\n \r\n \r\n 1\r\n
\ 2\r\n \r\n +\r\n
\ k\r\n ,\r\n Π\r\n \r\n \r\n
\ )\r\n \r\n
\ \r\n \r\n ∈\r\n \r\n (\r\n 2\r\n π\r\n i\r\n \r\n )\r\n \r\n
\ d\r\n ⋅\r\n k\r\n
\ \r\n \r\n \r\n Ω\r\n \r\n
\ (\r\n −\r\n
\ 1\r\n \r\n )\r\n
\ k\r\n \r\n \r\n
\ \r\n \r\n \\bf
Q\r\n \r\n (\r\n
\ Π\r\n )\r\n
\ ,\r\n \r\n \r\n
\ 1\r\n 2\r\n \r\n
\ +\r\n k\r\n \r\n critical\r\n ,\r\n \r\n
\ \\begin{equation*} L\\Big (\\frac
{1}{2}+k,\\Pi \\Big )\\;\\in \\;(2\\pi i)^{d\\cdot k}\\Omega _{(-1)^k}\\textrm
{\\bf Q}(\\Pi ),\\quad \\frac {1}{2}+k\\;\\text {critical}, \\end{equation*}\r\n
\ \r\n\r\n\r\n for certain automorphic
representations \r\n\r\n \r\n Π\r\n \\Pi\r\n
\ \r\n\r\n of a reductive group \r\n\r\n \r\n \r\n G\r\n
\ .\r\n \r\n G.\r\n
\ \r\n\r\n In this paper we discuss
the case \r\n\r\n \r\n \r\n G\r\n
\ =\r\n \r\n G\r\n L\r\n
\ \r\n (\r\n n\r\n
\ +\r\n 1\r\n )\r\n
\ ×\r\n \r\n
\ G\r\n L\r\n
\ \r\n (\r\n n\r\n
\ )\r\n .\r\n \r\n
\ G=\\mathrm {GL}(n+1)\\times
\\mathrm {GL}(n).\r\n \r\n\r\n
The case \r\n\r\n
\ \r\n \r\n G\r\n =\r\n
\ \r\n G\r\n
\ L\r\n \r\n (\r\n 2\r\n n\r\n
\ )\r\n \r\n G=\\mathrm {GL}(2n)\r\n \r\n\r\n
is discussed in part two. Our method is representation theoretic and relies on
the author’s recent results on global rational structures on automorphic representations.
We show that the above period relations are intimately related to the field of
definition of the global representation \r\n\r\n
\ \r\n Π\r\n
\ \\Pi\r\n \r\n\r\n
under consideration. The new period relations we prove are in accordance with
Deligne’s Conjecture on special values of \r\n\r\n \r\n
\ L\r\n L\r\n
\ \r\n\r\n-functions, and the author
expects this method to apply to other cases as well.
"
article_type: original
author:
- first_name: Fabian
full_name: Januszewski, Fabian
id: '81636'
last_name: Januszewski
orcid: 0000-0002-3184-237X
citation:
ama: "Januszewski F. On period relations for automorphic \U0001D43F-functions I.
Transactions of the American Mathematical Society. 2018;371(9):6547-6580.
doi:10.1090/tran/7527"
apa: "Januszewski, F. (2018). On period relations for automorphic \U0001D43F-functions
I. Transactions of the American Mathematical Society, 371(9), 6547–6580.
https://doi.org/10.1090/tran/7527"
bibtex: "@article{Januszewski_2018, title={On period relations for automorphic \U0001D43F-functions
I}, volume={371}, DOI={10.1090/tran/7527},
number={9}, journal={Transactions of the American Mathematical Society}, publisher={American
Mathematical Society (AMS)}, author={Januszewski, Fabian}, year={2018}, pages={6547–6580}
}"
chicago: "Januszewski, Fabian. “On Period Relations for Automorphic \U0001D43F-Functions
I.” Transactions of the American Mathematical Society 371, no. 9 (2018):
6547–80. https://doi.org/10.1090/tran/7527."
ieee: "F. Januszewski, “On period relations for automorphic \U0001D43F-functions
I,” Transactions of the American Mathematical Society, vol. 371, no. 9,
pp. 6547–6580, 2018, doi: 10.1090/tran/7527."
mla: "Januszewski, Fabian. “On Period Relations for Automorphic \U0001D43F-Functions
I.” Transactions of the American Mathematical Society, vol. 371, no. 9,
American Mathematical Society (AMS), 2018, pp. 6547–80, doi:10.1090/tran/7527."
short: F. Januszewski, Transactions of the American Mathematical Society 371 (2018)
6547–6580.
date_created: 2024-04-03T16:58:26Z
date_updated: 2024-04-03T17:26:38Z
doi: 10.1090/tran/7527
extern: '1'
intvolume: ' 371'
issue: '9'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
page: 6547-6580
publication: Transactions of the American Mathematical Society
publication_identifier:
issn:
- 0002-9947
- 1088-6850
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: "On period relations for automorphic \U0001D43F-functions I"
type: journal_article
user_id: '81636'
volume: 371
year: '2018'
...