---
_id: '60993'
abstract:
- lang: eng
text: "We determine the distribution of discriminants of wildly ramified elementary-abelian
extensions of local and global function fields in characteristic \r\n\r\n \r\n p\r\n p\r\n \r\n\r\n.
For local and rational function fields, we also give precise formulae for the
number of elementary-abelian extensions with a fixed discriminant divisor, which
describe a local-global principle.
"
author:
- first_name: Nicolas
full_name: Potthast, Nicolas
id: '63958'
last_name: Potthast
citation:
ama: Potthast N. On the asymptotics of elementary-abelian extensions of local and
global function fields. Transactions of the American Mathematical Society.
2026;379(1):289-340. doi:10.1090/tran/9509
apa: Potthast, N. (2026). On the asymptotics of elementary-abelian extensions of
local and global function fields. Transactions of the American Mathematical
Society, 379(1), 289–340. https://doi.org/10.1090/tran/9509
bibtex: '@article{Potthast_2026, title={On the asymptotics of elementary-abelian
extensions of local and global function fields}, volume={379}, DOI={10.1090/tran/9509},
number={1}, journal={Transactions of the American Mathematical Society}, publisher={American
Mathematical Society (AMS)}, author={Potthast, Nicolas}, year={2026}, pages={289–340}
}'
chicago: 'Potthast, Nicolas. “On the Asymptotics of Elementary-Abelian Extensions
of Local and Global Function Fields.” Transactions of the American Mathematical
Society 379, no. 1 (2026): 289–340. https://doi.org/10.1090/tran/9509.'
ieee: 'N. Potthast, “On the asymptotics of elementary-abelian extensions of local
and global function fields,” Transactions of the American Mathematical Society,
vol. 379, no. 1, pp. 289–340, 2026, doi: 10.1090/tran/9509.'
mla: Potthast, Nicolas. “On the Asymptotics of Elementary-Abelian Extensions of
Local and Global Function Fields.” Transactions of the American Mathematical
Society, vol. 379, no. 1, American Mathematical Society (AMS), 2026, pp. 289–340,
doi:10.1090/tran/9509.
short: N. Potthast, Transactions of the American Mathematical Society 379 (2026)
289–340.
date_created: 2025-08-25T12:37:47Z
date_updated: 2025-12-12T08:40:53Z
doi: 10.1090/tran/9509
intvolume: ' 379'
issue: '1'
language:
- iso: eng
page: 289 - 340
publication: Transactions of the American Mathematical Society
publication_identifier:
issn:
- 1088-6850
- 0002-9947
publication_status: published
publisher: American Mathematical Society (AMS)
status: public
title: On the asymptotics of elementary-abelian extensions of local and global function
fields
type: journal_article
user_id: '63958'
volume: 379
year: '2026'
...
---
_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'
...
---
_id: '38032'
author:
- first_name: Margit
full_name: Rösler, Margit
id: '37390'
last_name: Rösler
- first_name: Michael
full_name: Voit, Michael
last_name: Voit
citation:
ama: Rösler M, Voit M. Integral representation and sharp asymptotic results for
some Heckman-Opdam hypergeometric functions of type BC. Transactions of the
American Mathematical Society. 2016;368(8):6005-6032. doi:10.48550/ARXIV.1402.5793
apa: Rösler, M., & Voit, M. (2016). Integral representation and sharp asymptotic
results for some Heckman-Opdam hypergeometric functions of type BC. Transactions
of the American Mathematical Society, 368(8), 6005–6032. https://doi.org/10.48550/ARXIV.1402.5793
bibtex: '@article{Rösler_Voit_2016, title={Integral representation and sharp asymptotic
results for some Heckman-Opdam hypergeometric functions of type BC}, volume={368},
DOI={10.48550/ARXIV.1402.5793},
number={8}, journal={Transactions of the American Mathematical Society}, publisher={
American Mathematical Society}, author={Rösler, Margit and Voit, Michael}, year={2016},
pages={6005–6032} }'
chicago: 'Rösler, Margit, and Michael Voit. “Integral Representation and Sharp Asymptotic
Results for Some Heckman-Opdam Hypergeometric Functions of Type BC.” Transactions
of the American Mathematical Society 368, no. 8 (2016): 6005–32. https://doi.org/10.48550/ARXIV.1402.5793.'
ieee: 'M. Rösler and M. Voit, “Integral representation and sharp asymptotic results
for some Heckman-Opdam hypergeometric functions of type BC,” Transactions of
the American Mathematical Society, vol. 368, no. 8, pp. 6005–6032, 2016, doi:
10.48550/ARXIV.1402.5793.'
mla: Rösler, Margit, and Michael Voit. “Integral Representation and Sharp Asymptotic
Results for Some Heckman-Opdam Hypergeometric Functions of Type BC.” Transactions
of the American Mathematical Society, vol. 368, no. 8, American Mathematical
Society, 2016, pp. 6005–32, doi:10.48550/ARXIV.1402.5793.
short: M. Rösler, M. Voit, Transactions of the American Mathematical Society 368
(2016) 6005–6032.
date_created: 2023-01-23T08:09:20Z
date_updated: 2023-01-24T22:15:46Z
department:
- _id: '555'
doi: 10.48550/ARXIV.1402.5793
intvolume: ' 368'
issue: '8'
language:
- iso: eng
page: 6005-6032
publication: Transactions of the American Mathematical Society
publication_identifier:
issn:
- 1088-6850
publication_status: published
publisher: ' American Mathematical Society'
status: public
title: Integral representation and sharp asymptotic results for some Heckman-Opdam
hypergeometric functions of type BC
type: journal_article
user_id: '37390'
volume: 368
year: '2016'
...