Rational versus transcendental points on analytic Riemann surfaces
Let ( X , L ) be a polarized variety over a number field K . We suppose that L is an hermitian line bundle. Let M be a non compact Riemann Surface and U ⊂ M be a relatively compact open set. Let φ : M → X ( C ) be a holomorphic map. For every positive real number T , let A U ( T ) be the cardinalit...
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| Veröffentlicht in: | Manuscripta mathematica 2022-09, Vol.169 (1-2), p.77-105 |
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| creator | Gasbarri, Carlo |
| description | Let (
X
,
L
) be a polarized variety over a number field
K
. We suppose that
L
is an hermitian line bundle. Let
M
be a non compact Riemann Surface and
U
⊂
M
be a relatively compact open set. Let
φ
:
M
→
X
(
C
)
be a holomorphic map. For every positive real number
T
, let
A
U
(
T
)
be the cardinality of the set of
z
∈
U
such that
φ
(
z
)
∈
X
(
K
)
and
h
L
(
φ
(
z
)
)
≤
T
. After a revisitation of the proof of the sub exponential bound for
A
U
(
T
)
, obtained by Bombieri and Pila, we show that there are intervals of the reals such that for
T
in these intervals,
A
U
(
T
)
is upper bounded by a polynomial in
T
. We then introduce subsets of type
S
with respect of
φ
. These are compact subsets of
M
for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if
M
contains a subset of type
S
, then,
for every value of
T
the number
A
U
(
T
)
is bounded by a polynomial in
T
. As a consequence, we show that if
M
is a smooth leaf of an algebraic foliation in curves defined over
K
then
A
U
(
T
)
is bounded by a polynomial in
T
. Let
S
(
X
) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that
φ
-
1
(
S
(
X
)
)
≠
∅
if and only if
φ
-
1
(
S
(
X
)
)
is full for the Lebesgue measure on
M
. |
| doi_str_mv | 10.1007/s00229-021-01324-4 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_hal_p</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_03513365v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2699480845</sourcerecordid><originalsourceid>FETCH-LOGICAL-c348t-db05fa2ccc9ae27153845b9b45047831e4bfe6d311c932da9ca98e9a75f5e1413</originalsourceid><addsrcrecordid>eNp9kE9LAzEQxYMoWKtfwNOCJw_RTP7sbo61qBUKQtFzyGazuqXN1sxuwW9v6orePA28-b03zCPkEtgNMFbcImOca8o4UAaCSyqPyASk4BSKUh2TSdorynOAU3KGuGYsLQsxIXcr27ddsJts7yMOmPXRBnQ-1D70Sd11begx60JmE_TZty5btX5rQ8hwiI11Hs_JSWM36C9-5pS8Pty_zBd0-fz4NJ8tqROy7GldMdVY7pzT1vMClCilqnQlFZNFKcDLqvF5LQCcFry22lldem0L1SgPEsSUXI-573ZjdrHd2vhpOtuaxWxpDhoTCoTI1f7AXo3sLnYfg8ferLshpg_Q8FxrWbJ0PFF8pFzsEKNvfmOBmUOvZuzVpF7Nd69GJpMYTZjg8ObjX_Q_ri_LwHoH</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2699480845</pqid></control><display><type>article</type><title>Rational versus transcendental points on analytic Riemann surfaces</title><source>SpringerLink Journals - AutoHoldings</source><creator>Gasbarri, Carlo</creator><creatorcontrib>Gasbarri, Carlo</creatorcontrib><description>Let (
X
,
L
) be a polarized variety over a number field
K
. We suppose that
L
is an hermitian line bundle. Let
M
be a non compact Riemann Surface and
U
⊂
M
be a relatively compact open set. Let
φ
:
M
→
X
(
C
)
be a holomorphic map. For every positive real number
T
, let
A
U
(
T
)
be the cardinality of the set of
z
∈
U
such that
φ
(
z
)
∈
X
(
K
)
and
h
L
(
φ
(
z
)
)
≤
T
. After a revisitation of the proof of the sub exponential bound for
A
U
(
T
)
, obtained by Bombieri and Pila, we show that there are intervals of the reals such that for
T
in these intervals,
A
U
(
T
)
is upper bounded by a polynomial in
T
. We then introduce subsets of type
S
with respect of
φ
. These are compact subsets of
M
for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if
M
contains a subset of type
S
, then,
for every value of
T
the number
A
U
(
T
)
is bounded by a polynomial in
T
. As a consequence, we show that if
M
is a smooth leaf of an algebraic foliation in curves defined over
K
then
A
U
(
T
)
is bounded by a polynomial in
T
. Let
S
(
X
) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that
φ
-
1
(
S
(
X
)
)
≠
∅
if and only if
φ
-
1
(
S
(
X
)
)
is full for the Lebesgue measure on
M
.</description><identifier>ISSN: 0025-2611</identifier><identifier>EISSN: 1432-1785</identifier><identifier>DOI: 10.1007/s00229-021-01324-4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebra ; Algebraic Geometry ; Calculus of Variations and Optimal Control; Optimization ; Fields (mathematics) ; Geometry ; Intervals ; Lie Groups ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Number Theory ; Polynomials ; Real numbers ; Riemann surfaces ; Set theory ; Topological Groups</subject><ispartof>Manuscripta mathematica, 2022-09, Vol.169 (1-2), p.77-105</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c348t-db05fa2ccc9ae27153845b9b45047831e4bfe6d311c932da9ca98e9a75f5e1413</cites><orcidid>0000-0003-4709-8511</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/s00229-021-01324-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00229-021-01324-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27922,27923,41486,42555,51317</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03513365$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Gasbarri, Carlo</creatorcontrib><title>Rational versus transcendental points on analytic Riemann surfaces</title><title>Manuscripta mathematica</title><addtitle>manuscripta math</addtitle><description>Let (
X
,
L
) be a polarized variety over a number field
K
. We suppose that
L
is an hermitian line bundle. Let
M
be a non compact Riemann Surface and
U
⊂
M
be a relatively compact open set. Let
φ
:
M
→
X
(
C
)
be a holomorphic map. For every positive real number
T
, let
A
U
(
T
)
be the cardinality of the set of
z
∈
U
such that
φ
(
z
)
∈
X
(
K
)
and
h
L
(
φ
(
z
)
)
≤
T
. After a revisitation of the proof of the sub exponential bound for
A
U
(
T
)
, obtained by Bombieri and Pila, we show that there are intervals of the reals such that for
T
in these intervals,
A
U
(
T
)
is upper bounded by a polynomial in
T
. We then introduce subsets of type
S
with respect of
φ
. These are compact subsets of
M
for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if
M
contains a subset of type
S
, then,
for every value of
T
the number
A
U
(
T
)
is bounded by a polynomial in
T
. As a consequence, we show that if
M
is a smooth leaf of an algebraic foliation in curves defined over
K
then
A
U
(
T
)
is bounded by a polynomial in
T
. Let
S
(
X
) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that
φ
-
1
(
S
(
X
)
)
≠
∅
if and only if
φ
-
1
(
S
(
X
)
)
is full for the Lebesgue measure on
M
.</description><subject>Algebra</subject><subject>Algebraic Geometry</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Fields (mathematics)</subject><subject>Geometry</subject><subject>Intervals</subject><subject>Lie Groups</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Polynomials</subject><subject>Real numbers</subject><subject>Riemann surfaces</subject><subject>Set theory</subject><subject>Topological Groups</subject><issn>0025-2611</issn><issn>1432-1785</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LAzEQxYMoWKtfwNOCJw_RTP7sbo61qBUKQtFzyGazuqXN1sxuwW9v6orePA28-b03zCPkEtgNMFbcImOca8o4UAaCSyqPyASk4BSKUh2TSdorynOAU3KGuGYsLQsxIXcr27ddsJts7yMOmPXRBnQ-1D70Sd11begx60JmE_TZty5btX5rQ8hwiI11Hs_JSWM36C9-5pS8Pty_zBd0-fz4NJ8tqROy7GldMdVY7pzT1vMClCilqnQlFZNFKcDLqvF5LQCcFry22lldem0L1SgPEsSUXI-573ZjdrHd2vhpOtuaxWxpDhoTCoTI1f7AXo3sLnYfg8ferLshpg_Q8FxrWbJ0PFF8pFzsEKNvfmOBmUOvZuzVpF7Nd69GJpMYTZjg8ObjX_Q_ri_LwHoH</recordid><startdate>20220901</startdate><enddate>20220901</enddate><creator>Gasbarri, Carlo</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0003-4709-8511</orcidid></search><sort><creationdate>20220901</creationdate><title>Rational versus transcendental points on analytic Riemann surfaces</title><author>Gasbarri, Carlo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c348t-db05fa2ccc9ae27153845b9b45047831e4bfe6d311c932da9ca98e9a75f5e1413</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Algebraic Geometry</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Fields (mathematics)</topic><topic>Geometry</topic><topic>Intervals</topic><topic>Lie Groups</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Polynomials</topic><topic>Real numbers</topic><topic>Riemann surfaces</topic><topic>Set theory</topic><topic>Topological Groups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gasbarri, Carlo</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Manuscripta mathematica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gasbarri, Carlo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Rational versus transcendental points on analytic Riemann surfaces</atitle><jtitle>Manuscripta mathematica</jtitle><stitle>manuscripta math</stitle><date>2022-09-01</date><risdate>2022</risdate><volume>169</volume><issue>1-2</issue><spage>77</spage><epage>105</epage><pages>77-105</pages><issn>0025-2611</issn><eissn>1432-1785</eissn><abstract>Let (
X
,
L
) be a polarized variety over a number field
K
. We suppose that
L
is an hermitian line bundle. Let
M
be a non compact Riemann Surface and
U
⊂
M
be a relatively compact open set. Let
φ
:
M
→
X
(
C
)
be a holomorphic map. For every positive real number
T
, let
A
U
(
T
)
be the cardinality of the set of
z
∈
U
such that
φ
(
z
)
∈
X
(
K
)
and
h
L
(
φ
(
z
)
)
≤
T
. After a revisitation of the proof of the sub exponential bound for
A
U
(
T
)
, obtained by Bombieri and Pila, we show that there are intervals of the reals such that for
T
in these intervals,
A
U
(
T
)
is upper bounded by a polynomial in
T
. We then introduce subsets of type
S
with respect of
φ
. These are compact subsets of
M
for which an inequality similar to Liouville inequality on algebraic points holds. We show that, if
M
contains a subset of type
S
, then,
for every value of
T
the number
A
U
(
T
)
is bounded by a polynomial in
T
. As a consequence, we show that if
M
is a smooth leaf of an algebraic foliation in curves defined over
K
then
A
U
(
T
)
is bounded by a polynomial in
T
. Let
S
(
X
) be the subset (full for the Lebesgue measure) of points which verify some kind of Liouville inequalities. In the second part we prove that
φ
-
1
(
S
(
X
)
)
≠
∅
if and only if
φ
-
1
(
S
(
X
)
)
is full for the Lebesgue measure on
M
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00229-021-01324-4</doi><tpages>29</tpages><orcidid>https://orcid.org/0000-0003-4709-8511</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-2611 |
| ispartof | Manuscripta mathematica, 2022-09, Vol.169 (1-2), p.77-105 |
| issn | 0025-2611 1432-1785 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_03513365v1 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Algebra Algebraic Geometry Calculus of Variations and Optimal Control Optimization Fields (mathematics) Geometry Intervals Lie Groups Mathematical analysis Mathematics Mathematics and Statistics Number Theory Polynomials Real numbers Riemann surfaces Set theory Topological Groups |
| title | Rational versus transcendental points on analytic Riemann surfaces |
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