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On elements of prescribed norm in maximal orders of a quaternion algebra
Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p , $\mathcal {O}$...
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| Published in: | Canadian journal of mathematics 2024-11, p.1-28 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| container_end_page | 28 |
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| container_start_page | 1 |
| container_title | Canadian journal of mathematics |
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| creator | Goren, Eyal Z. Love, Jonathan R. |
| description | Let
$\mathcal {O}$
be a maximal order in the quaternion algebra over
$\mathbb Q$
ramified at
p
and
$\infty $
. We prove two theorems that allow us to recover the structure of
$\mathcal {O}$
from limited information. The first says that for any infinite set
S
of integers coprime to
p
,
$\mathcal {O}$
is spanned as a
${\mathbb {Z}}$
-module by elements with norm in
S
. The second says that
$\mathcal {O}$
is determined up to isomorphism by its theta function. |
| doi_str_mv | 10.4153/S0008414X24000592 |
| format | article |
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$\mathcal {O}$
be a maximal order in the quaternion algebra over
$\mathbb Q$
ramified at
p
and
$\infty $
. We prove two theorems that allow us to recover the structure of
$\mathcal {O}$
from limited information. The first says that for any infinite set
S
of integers coprime to
p
,
$\mathcal {O}$
is spanned as a
${\mathbb {Z}}$
-module by elements with norm in
S
. The second says that
$\mathcal {O}$
is determined up to isomorphism by its theta function.</description><identifier>ISSN: 0008-414X</identifier><identifier>EISSN: 1496-4279</identifier><identifier>DOI: 10.4153/S0008414X24000592</identifier><language>eng</language><ispartof>Canadian journal of mathematics, 2024-11, p.1-28</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c170t-76b42d338c6f92d2d3722707e61f7bd5c8d154edc4282b63791e9b68d449f2373</cites><orcidid>0000-0002-8167-3404 ; 0000-0002-6283-9931</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Goren, Eyal Z.</creatorcontrib><creatorcontrib>Love, Jonathan R.</creatorcontrib><title>On elements of prescribed norm in maximal orders of a quaternion algebra</title><title>Canadian journal of mathematics</title><description>Let
$\mathcal {O}$
be a maximal order in the quaternion algebra over
$\mathbb Q$
ramified at
p
and
$\infty $
. We prove two theorems that allow us to recover the structure of
$\mathcal {O}$
from limited information. The first says that for any infinite set
S
of integers coprime to
p
,
$\mathcal {O}$
is spanned as a
${\mathbb {Z}}$
-module by elements with norm in
S
. The second says that
$\mathcal {O}$
is determined up to isomorphism by its theta function.</description><issn>0008-414X</issn><issn>1496-4279</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNplUM1KxDAYDKJgXX0Ab3mBar4kTZqjLOoKC3tQwVvJzxeptOmaVNC3t6vePM0MMwzDEHIJ7EpCI64fGWOtBPnC5cIaw49IBdKoWnJtjkl1sOuDf0rOSnlbpFANVGSzSxQHHDHNhU6R7jMWn3uHgaYpj7RPdLSf_WgHOuWA-Sdk6fuHnTGnfkrUDq_osj0nJ9EOBS_-cEWe726f1pt6u7t_WN9saw-azbVWTvIgROtVNDwsVHOumUYFUbvQ-DZAIzF4yVvulNAG0DjVBilN5EKLFYHfXp-nUjLGbp-XefmrA9Ydruj-XSG-AS7vULo</recordid><startdate>20241111</startdate><enddate>20241111</enddate><creator>Goren, Eyal Z.</creator><creator>Love, Jonathan R.</creator><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-8167-3404</orcidid><orcidid>https://orcid.org/0000-0002-6283-9931</orcidid></search><sort><creationdate>20241111</creationdate><title>On elements of prescribed norm in maximal orders of a quaternion algebra</title><author>Goren, Eyal Z. ; Love, Jonathan R.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c170t-76b42d338c6f92d2d3722707e61f7bd5c8d154edc4282b63791e9b68d449f2373</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Goren, Eyal Z.</creatorcontrib><creatorcontrib>Love, Jonathan R.</creatorcontrib><collection>CrossRef</collection><jtitle>Canadian journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Goren, Eyal Z.</au><au>Love, Jonathan R.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On elements of prescribed norm in maximal orders of a quaternion algebra</atitle><jtitle>Canadian journal of mathematics</jtitle><date>2024-11-11</date><risdate>2024</risdate><spage>1</spage><epage>28</epage><pages>1-28</pages><issn>0008-414X</issn><eissn>1496-4279</eissn><abstract>Let
$\mathcal {O}$
be a maximal order in the quaternion algebra over
$\mathbb Q$
ramified at
p
and
$\infty $
. We prove two theorems that allow us to recover the structure of
$\mathcal {O}$
from limited information. The first says that for any infinite set
S
of integers coprime to
p
,
$\mathcal {O}$
is spanned as a
${\mathbb {Z}}$
-module by elements with norm in
S
. The second says that
$\mathcal {O}$
is determined up to isomorphism by its theta function.</abstract><doi>10.4153/S0008414X24000592</doi><tpages>28</tpages><orcidid>https://orcid.org/0000-0002-8167-3404</orcidid><orcidid>https://orcid.org/0000-0002-6283-9931</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0008-414X |
| ispartof | Canadian journal of mathematics, 2024-11, p.1-28 |
| issn | 0008-414X 1496-4279 |
| language | eng |
| recordid | cdi_crossref_primary_10_4153_S0008414X24000592 |
| source | Cambridge University Press |
| title | On elements of prescribed norm in maximal orders of a quaternion algebra |
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