Patterns of primes in the Sato–Tate conjecture
Fix a non-CM elliptic curve E / Q , and let a E ( p ) = p + 1 - # E ( F p ) denote the trace of Frobenius at p . The Sato–Tate conjecture gives the limiting distribution μ ST of a E ( p ) / ( 2 p ) within [ - 1 , 1 ] . We establish bounded gaps for primes in the context of this distribution. More pr...
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| Veröffentlicht in: | Research in number theory 2020, Vol.6 (1), Article 9 |
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| creator | Gillman, Nate Kural, Michael Pascadi, Alexandru Peng, Junyao Sah, Ashwin |
| description | Fix a non-CM elliptic curve
E
/
Q
, and let
a
E
(
p
)
=
p
+
1
-
#
E
(
F
p
)
denote the trace of Frobenius at
p
. The Sato–Tate conjecture gives the limiting distribution
μ
ST
of
a
E
(
p
)
/
(
2
p
)
within
[
-
1
,
1
]
. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval
I
⊆
[
-
1
,
1
]
, let
p
I
,
n
denote the
n
th prime such that
a
E
(
p
)
/
(
2
p
)
∈
I
. We show
lim inf
n
→
∞
(
p
I
,
n
+
m
-
p
I
,
n
)
<
∞
for all
m
≥
1
for “most” intervals, and in particular, for all
I
with
μ
ST
(
I
)
≥
0.36
. Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes. |
| doi_str_mv | 10.1007/s40993-019-0184-8 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2333963210</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2333963210</sourcerecordid><originalsourceid>FETCH-LOGICAL-c268t-562578cc21efd348557016dffddab350a013070b8e57b922b9bc81060183d1863</originalsourceid><addsrcrecordid>eNp1UM1KxDAQDqLgsu4DeCt4rk5mmjQ5yuIfLCi4nkOaprqLtmuSHrz5Dr6hT2KWCp48DDMw38_Mx9gph3MOUF_ECrSmErjOpapSHbAZkqRSCyEO8ywQ80bCMVvEuAXIM1WIOGPwYFPyoY_F0BW7sHnzsdj0RXrxxaNNw_fn19omX7ih33qXxuBP2FFnX6Nf_PY5e7q-Wi9vy9X9zd3yclU6lCqVQqKolXPIfddSpYSos2nbdW1rGxJggRPU0Cgv6kYjNrpxioPM91PLlaQ5O5t0d2F4H31MZjuMoc-WBolIS0IOGcUnlAtDjMF3Zv-DDR-Gg9lnY6ZsTM7G7LMxKnNw4sSM7Z99-FP-n_QD3QVk7w</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2333963210</pqid></control><display><type>article</type><title>Patterns of primes in the Sato–Tate conjecture</title><source>SpringerLink Journals</source><creator>Gillman, Nate ; Kural, Michael ; Pascadi, Alexandru ; Peng, Junyao ; Sah, Ashwin</creator><creatorcontrib>Gillman, Nate ; Kural, Michael ; Pascadi, Alexandru ; Peng, Junyao ; Sah, Ashwin</creatorcontrib><description>Fix a non-CM elliptic curve
E
/
Q
, and let
a
E
(
p
)
=
p
+
1
-
#
E
(
F
p
)
denote the trace of Frobenius at
p
. The Sato–Tate conjecture gives the limiting distribution
μ
ST
of
a
E
(
p
)
/
(
2
p
)
within
[
-
1
,
1
]
. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval
I
⊆
[
-
1
,
1
]
, let
p
I
,
n
denote the
n
th prime such that
a
E
(
p
)
/
(
2
p
)
∈
I
. We show
lim inf
n
→
∞
(
p
I
,
n
+
m
-
p
I
,
n
)
<
∞
for all
m
≥
1
for “most” intervals, and in particular, for all
I
with
μ
ST
(
I
)
≥
0.36
. Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes.</description><identifier>ISSN: 2522-0160</identifier><identifier>EISSN: 2363-9555</identifier><identifier>DOI: 10.1007/s40993-019-0184-8</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Curves ; Mathematics ; Mathematics and Statistics ; Number Theory ; Theorems</subject><ispartof>Research in number theory, 2020, Vol.6 (1), Article 9</ispartof><rights>Springer Nature Switzerland AG 2019</rights><rights>2019© Springer Nature Switzerland AG 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c268t-562578cc21efd348557016dffddab350a013070b8e57b922b9bc81060183d1863</cites><orcidid>0000-0001-9286-9165</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s40993-019-0184-8$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40993-019-0184-8$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Gillman, Nate</creatorcontrib><creatorcontrib>Kural, Michael</creatorcontrib><creatorcontrib>Pascadi, Alexandru</creatorcontrib><creatorcontrib>Peng, Junyao</creatorcontrib><creatorcontrib>Sah, Ashwin</creatorcontrib><title>Patterns of primes in the Sato–Tate conjecture</title><title>Research in number theory</title><addtitle>Res. number theory</addtitle><description>Fix a non-CM elliptic curve
E
/
Q
, and let
a
E
(
p
)
=
p
+
1
-
#
E
(
F
p
)
denote the trace of Frobenius at
p
. The Sato–Tate conjecture gives the limiting distribution
μ
ST
of
a
E
(
p
)
/
(
2
p
)
within
[
-
1
,
1
]
. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval
I
⊆
[
-
1
,
1
]
, let
p
I
,
n
denote the
n
th prime such that
a
E
(
p
)
/
(
2
p
)
∈
I
. We show
lim inf
n
→
∞
(
p
I
,
n
+
m
-
p
I
,
n
)
<
∞
for all
m
≥
1
for “most” intervals, and in particular, for all
I
with
μ
ST
(
I
)
≥
0.36
. Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes.</description><subject>Curves</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Theorems</subject><issn>2522-0160</issn><issn>2363-9555</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp1UM1KxDAQDqLgsu4DeCt4rk5mmjQ5yuIfLCi4nkOaprqLtmuSHrz5Dr6hT2KWCp48DDMw38_Mx9gph3MOUF_ECrSmErjOpapSHbAZkqRSCyEO8ywQ80bCMVvEuAXIM1WIOGPwYFPyoY_F0BW7sHnzsdj0RXrxxaNNw_fn19omX7ih33qXxuBP2FFnX6Nf_PY5e7q-Wi9vy9X9zd3yclU6lCqVQqKolXPIfddSpYSos2nbdW1rGxJggRPU0Cgv6kYjNrpxioPM91PLlaQ5O5t0d2F4H31MZjuMoc-WBolIS0IOGcUnlAtDjMF3Zv-DDR-Gg9lnY6ZsTM7G7LMxKnNw4sSM7Z99-FP-n_QD3QVk7w</recordid><startdate>2020</startdate><enddate>2020</enddate><creator>Gillman, Nate</creator><creator>Kural, Michael</creator><creator>Pascadi, Alexandru</creator><creator>Peng, Junyao</creator><creator>Sah, Ashwin</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-9286-9165</orcidid></search><sort><creationdate>2020</creationdate><title>Patterns of primes in the Sato–Tate conjecture</title><author>Gillman, Nate ; Kural, Michael ; Pascadi, Alexandru ; Peng, Junyao ; Sah, Ashwin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c268t-562578cc21efd348557016dffddab350a013070b8e57b922b9bc81060183d1863</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Curves</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gillman, Nate</creatorcontrib><creatorcontrib>Kural, Michael</creatorcontrib><creatorcontrib>Pascadi, Alexandru</creatorcontrib><creatorcontrib>Peng, Junyao</creatorcontrib><creatorcontrib>Sah, Ashwin</creatorcontrib><collection>CrossRef</collection><jtitle>Research in number theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gillman, Nate</au><au>Kural, Michael</au><au>Pascadi, Alexandru</au><au>Peng, Junyao</au><au>Sah, Ashwin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Patterns of primes in the Sato–Tate conjecture</atitle><jtitle>Research in number theory</jtitle><stitle>Res. number theory</stitle><date>2020</date><risdate>2020</risdate><volume>6</volume><issue>1</issue><artnum>9</artnum><issn>2522-0160</issn><eissn>2363-9555</eissn><abstract>Fix a non-CM elliptic curve
E
/
Q
, and let
a
E
(
p
)
=
p
+
1
-
#
E
(
F
p
)
denote the trace of Frobenius at
p
. The Sato–Tate conjecture gives the limiting distribution
μ
ST
of
a
E
(
p
)
/
(
2
p
)
within
[
-
1
,
1
]
. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval
I
⊆
[
-
1
,
1
]
, let
p
I
,
n
denote the
n
th prime such that
a
E
(
p
)
/
(
2
p
)
∈
I
. We show
lim inf
n
→
∞
(
p
I
,
n
+
m
-
p
I
,
n
)
<
∞
for all
m
≥
1
for “most” intervals, and in particular, for all
I
with
μ
ST
(
I
)
≥
0.36
. Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s40993-019-0184-8</doi><orcidid>https://orcid.org/0000-0001-9286-9165</orcidid></addata></record> |
| fulltext | fulltext |
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| ispartof | Research in number theory, 2020, Vol.6 (1), Article 9 |
| issn | 2522-0160 2363-9555 |
| language | eng |
| recordid | cdi_proquest_journals_2333963210 |
| source | SpringerLink Journals |
| subjects | Curves Mathematics Mathematics and Statistics Number Theory Theorems |
| title | Patterns of primes in the Sato–Tate conjecture |
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