Algebraic points on a curve with small degree












2














Let $d geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $mathbb{Q}$.



Let $Y$ be an algebraic curve defined over the rationals and has genus $g geq 2$. It is a well-known theorem of Faltings that asserts that for any number field $K$, the set of $K$-points of $Y$ is finite. However, if $Y$ is given by a plane model of the form $F(x,y,z) = 0$ for some homogeneous polynomial $F$ with rational coefficients and degree $d$, then certainly $Y$ has infinitely many points over $K_d$. Indeed, one can choose $x,y$ to be any two rational numbers, and the corresponding equation in $z$ will surely have a solution over $K_d$.



Sometimes $Y$ has infinitely many points over $K_d$ for a much smaller $d$. For instance if $Y$ is hyperelliptic, then $Y$ always has infinitely many points over $K_2$.



The gonality of $Y$, given as a plane curve, is the smallest positive integer $k$ for which $Y$ admits a degree $k$, non-constant map to the projective line $mathbb{P}^1$. Hyperelliptic curves have gonality equal to two, which explains the behaviour above. In general, if $m$ is the gonality of $Y$, then $Y(K_m)$ will be infinite.



In general, if $Y$ is given as a plane curve by a homogeneous polynomial $F(x,y,z) = 0$ of gonality $m$, do we expect $Y(K_s)$ to be finite or infinite for $s < m$?



Edit: thanks to Jason Starr for pointing out the notion of gonality.






nt.number-theory arithmetic-geometry







share|cite|improve this question
























  • Are you asking about the gonality of a plane curve?
    – Jason Starr
    4 hours ago










  • I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
    – Stanley Yao Xiao
    3 hours ago










  • It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
    – Xarles
    2 hours ago
















2














Let $d geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $mathbb{Q}$.



Let $Y$ be an algebraic curve defined over the rationals and has genus $g geq 2$. It is a well-known theorem of Faltings that asserts that for any number field $K$, the set of $K$-points of $Y$ is finite. However, if $Y$ is given by a plane model of the form $F(x,y,z) = 0$ for some homogeneous polynomial $F$ with rational coefficients and degree $d$, then certainly $Y$ has infinitely many points over $K_d$. Indeed, one can choose $x,y$ to be any two rational numbers, and the corresponding equation in $z$ will surely have a solution over $K_d$.



Sometimes $Y$ has infinitely many points over $K_d$ for a much smaller $d$. For instance if $Y$ is hyperelliptic, then $Y$ always has infinitely many points over $K_2$.



The gonality of $Y$, given as a plane curve, is the smallest positive integer $k$ for which $Y$ admits a degree $k$, non-constant map to the projective line $mathbb{P}^1$. Hyperelliptic curves have gonality equal to two, which explains the behaviour above. In general, if $m$ is the gonality of $Y$, then $Y(K_m)$ will be infinite.



In general, if $Y$ is given as a plane curve by a homogeneous polynomial $F(x,y,z) = 0$ of gonality $m$, do we expect $Y(K_s)$ to be finite or infinite for $s < m$?



Edit: thanks to Jason Starr for pointing out the notion of gonality.






nt.number-theory arithmetic-geometry







share|cite|improve this question
























  • Are you asking about the gonality of a plane curve?
    – Jason Starr
    4 hours ago










  • I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
    – Stanley Yao Xiao
    3 hours ago










  • It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
    – Xarles
    2 hours ago














2












2








2







Let $d geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $mathbb{Q}$.



Let $Y$ be an algebraic curve defined over the rationals and has genus $g geq 2$. It is a well-known theorem of Faltings that asserts that for any number field $K$, the set of $K$-points of $Y$ is finite. However, if $Y$ is given by a plane model of the form $F(x,y,z) = 0$ for some homogeneous polynomial $F$ with rational coefficients and degree $d$, then certainly $Y$ has infinitely many points over $K_d$. Indeed, one can choose $x,y$ to be any two rational numbers, and the corresponding equation in $z$ will surely have a solution over $K_d$.



Sometimes $Y$ has infinitely many points over $K_d$ for a much smaller $d$. For instance if $Y$ is hyperelliptic, then $Y$ always has infinitely many points over $K_2$.



The gonality of $Y$, given as a plane curve, is the smallest positive integer $k$ for which $Y$ admits a degree $k$, non-constant map to the projective line $mathbb{P}^1$. Hyperelliptic curves have gonality equal to two, which explains the behaviour above. In general, if $m$ is the gonality of $Y$, then $Y(K_m)$ will be infinite.



In general, if $Y$ is given as a plane curve by a homogeneous polynomial $F(x,y,z) = 0$ of gonality $m$, do we expect $Y(K_s)$ to be finite or infinite for $s < m$?



Edit: thanks to Jason Starr for pointing out the notion of gonality.






nt.number-theory arithmetic-geometry







share|cite|improve this question















Let $d geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $mathbb{Q}$.



Let $Y$ be an algebraic curve defined over the rationals and has genus $g geq 2$. It is a well-known theorem of Faltings that asserts that for any number field $K$, the set of $K$-points of $Y$ is finite. However, if $Y$ is given by a plane model of the form $F(x,y,z) = 0$ for some homogeneous polynomial $F$ with rational coefficients and degree $d$, then certainly $Y$ has infinitely many points over $K_d$. Indeed, one can choose $x,y$ to be any two rational numbers, and the corresponding equation in $z$ will surely have a solution over $K_d$.



Sometimes $Y$ has infinitely many points over $K_d$ for a much smaller $d$. For instance if $Y$ is hyperelliptic, then $Y$ always has infinitely many points over $K_2$.



The gonality of $Y$, given as a plane curve, is the smallest positive integer $k$ for which $Y$ admits a degree $k$, non-constant map to the projective line $mathbb{P}^1$. Hyperelliptic curves have gonality equal to two, which explains the behaviour above. In general, if $m$ is the gonality of $Y$, then $Y(K_m)$ will be infinite.



In general, if $Y$ is given as a plane curve by a homogeneous polynomial $F(x,y,z) = 0$ of gonality $m$, do we expect $Y(K_s)$ to be finite or infinite for $s < m$?



Edit: thanks to Jason Starr for pointing out the notion of gonality.






nt.number-theory arithmetic-geometry




nt.number-theory arithmetic-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago

























asked 4 hours ago









Stanley Yao Xiao

8,43642784




8,43642784












  • Are you asking about the gonality of a plane curve?
    – Jason Starr
    4 hours ago










  • I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
    – Stanley Yao Xiao
    3 hours ago










  • It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
    – Xarles
    2 hours ago


















  • Are you asking about the gonality of a plane curve?
    – Jason Starr
    4 hours ago










  • I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
    – Stanley Yao Xiao
    3 hours ago










  • It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
    – Xarles
    2 hours ago
















Are you asking about the gonality of a plane curve?
– Jason Starr
4 hours ago




Are you asking about the gonality of a plane curve?
– Jason Starr
4 hours ago












I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
– Stanley Yao Xiao
3 hours ago




I was not aware of the notion of gonality before; I will edit the question to reflect this new notion.
– Stanley Yao Xiao
3 hours ago












It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
– Xarles
2 hours ago




It depends also if it has a map of degree say $r$ to an elliptic curve of rank >0. For example bielliptic curves (that have a degree 2 map to an elliptic curve) have infinitely many degree 2 points over any field where some elliptic quotient has rank >0.
– Xarles
2 hours ago










1 Answer
1






active

oldest

votes


















4














In the first place, there's also the possibility that there is a map of degree $m$ from $Y$ to an elliptic curve $E$ such that $E(mathbb Q)$ is infinite, in which case $Y(K_m)$ will clearly be infinite. There's actually a bunch of literature on this subject, including for example:




  • MR1055774 Harris, Joe; Silverman, Joe; Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. (This deals with quadratic points, so in your notation, with $K_2$. The answer is that in order for $Y(K_2)$ to be infinite, the curve $Y$ must be hyperelliptic or admit a degree 2 map to an elliptic curve.)


  • MR11047 Abramovich, Dan; Harris, Joe; Abelian varieties and curves in $W_d(C)$. Compositio Math. 78 (1991), no. 2, 227–238. (This deals with the general case. The answer is a bit more complicated than just looking at gonality.)



In all cases, the results rely on Faltings' theorem describing subvarieties of $A(K)$ having having a Zariski dense set of $K$-rational points.






share|cite|improve this answer





















  • I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
    – Xarles
    2 hours ago











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320015%2falgebraic-points-on-a-curve-with-small-degree%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









4














In the first place, there's also the possibility that there is a map of degree $m$ from $Y$ to an elliptic curve $E$ such that $E(mathbb Q)$ is infinite, in which case $Y(K_m)$ will clearly be infinite. There's actually a bunch of literature on this subject, including for example:




  • MR1055774 Harris, Joe; Silverman, Joe; Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. (This deals with quadratic points, so in your notation, with $K_2$. The answer is that in order for $Y(K_2)$ to be infinite, the curve $Y$ must be hyperelliptic or admit a degree 2 map to an elliptic curve.)


  • MR11047 Abramovich, Dan; Harris, Joe; Abelian varieties and curves in $W_d(C)$. Compositio Math. 78 (1991), no. 2, 227–238. (This deals with the general case. The answer is a bit more complicated than just looking at gonality.)



In all cases, the results rely on Faltings' theorem describing subvarieties of $A(K)$ having having a Zariski dense set of $K$-rational points.






share|cite|improve this answer





















  • I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
    – Xarles
    2 hours ago
















4














In the first place, there's also the possibility that there is a map of degree $m$ from $Y$ to an elliptic curve $E$ such that $E(mathbb Q)$ is infinite, in which case $Y(K_m)$ will clearly be infinite. There's actually a bunch of literature on this subject, including for example:




  • MR1055774 Harris, Joe; Silverman, Joe; Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. (This deals with quadratic points, so in your notation, with $K_2$. The answer is that in order for $Y(K_2)$ to be infinite, the curve $Y$ must be hyperelliptic or admit a degree 2 map to an elliptic curve.)


  • MR11047 Abramovich, Dan; Harris, Joe; Abelian varieties and curves in $W_d(C)$. Compositio Math. 78 (1991), no. 2, 227–238. (This deals with the general case. The answer is a bit more complicated than just looking at gonality.)



In all cases, the results rely on Faltings' theorem describing subvarieties of $A(K)$ having having a Zariski dense set of $K$-rational points.






share|cite|improve this answer





















  • I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
    – Xarles
    2 hours ago














4












4








4






In the first place, there's also the possibility that there is a map of degree $m$ from $Y$ to an elliptic curve $E$ such that $E(mathbb Q)$ is infinite, in which case $Y(K_m)$ will clearly be infinite. There's actually a bunch of literature on this subject, including for example:




  • MR1055774 Harris, Joe; Silverman, Joe; Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. (This deals with quadratic points, so in your notation, with $K_2$. The answer is that in order for $Y(K_2)$ to be infinite, the curve $Y$ must be hyperelliptic or admit a degree 2 map to an elliptic curve.)


  • MR11047 Abramovich, Dan; Harris, Joe; Abelian varieties and curves in $W_d(C)$. Compositio Math. 78 (1991), no. 2, 227–238. (This deals with the general case. The answer is a bit more complicated than just looking at gonality.)



In all cases, the results rely on Faltings' theorem describing subvarieties of $A(K)$ having having a Zariski dense set of $K$-rational points.






share|cite|improve this answer












In the first place, there's also the possibility that there is a map of degree $m$ from $Y$ to an elliptic curve $E$ such that $E(mathbb Q)$ is infinite, in which case $Y(K_m)$ will clearly be infinite. There's actually a bunch of literature on this subject, including for example:




  • MR1055774 Harris, Joe; Silverman, Joe; Bielliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. (This deals with quadratic points, so in your notation, with $K_2$. The answer is that in order for $Y(K_2)$ to be infinite, the curve $Y$ must be hyperelliptic or admit a degree 2 map to an elliptic curve.)


  • MR11047 Abramovich, Dan; Harris, Joe; Abelian varieties and curves in $W_d(C)$. Compositio Math. 78 (1991), no. 2, 227–238. (This deals with the general case. The answer is a bit more complicated than just looking at gonality.)



In all cases, the results rely on Faltings' theorem describing subvarieties of $A(K)$ having having a Zariski dense set of $K$-rational points.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 hours ago









Joe Silverman

30.3k180157




30.3k180157












  • I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
    – Xarles
    2 hours ago


















  • I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
    – Xarles
    2 hours ago
















I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
– Xarles
2 hours ago




I think the OP asks about the points in what he call $K_2$, which is the composite of all the degree 2 extensions of $K$, not just the degree 2 points (in the usual sense of your paper with Harris). I guess there are (modular?) curves with infinitely many points in $K_2$ but finitely many quadratic points for $K=mathbb{Q}$.
– Xarles
2 hours ago


















draft saved

draft discarded




















































Thanks for contributing an answer to MathOverflow!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f320015%2falgebraic-points-on-a-curve-with-small-degree%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Eastern Orthodox Church

Image
Not to be confused with Oriental Orthodoxy, Eastern Christianity, or Eastern Catholic Churches. "Eastern Orthodoxy", "Orthodox Church", "Orthodox Catholic Church", and "Orthodox Christian Church" redirect here. For other uses, see Orthodox (disambiguation). This article may be too long to read and navigate comfortably . The readable prose size is 132 kilobytes. Please consider splitting content into sub-articles, condensing it, or adding or removing subheadings. ( January 2018 ) Part of a series on the Eastern Orthodox Church Mosaic of Christ Pantocrator, Hagia Sophia Overview Structure Theology (History of theology) Liturgy Church history Holy Mysteries View of salvation View of Mary View of icons Background Crucifixion / Resurrection / Ascension of Jesus Christianity Christian Church Apostolic succession Four Marks of the Church Orthodoxy Organization Autocephaly Patriarchate ...
Read more

Zagreb

Image
This article is about the Croatian capital city. For other uses, see Zagreb (disambiguation). City in City of Zagreb, Croatia Zagreb City Grad Zagreb City of Zagreb Clockwise, from top: St. Mark's Square, Croatian State Archives, Zagreb Tram, Art Pavilion, Cibona & HOTO towers and Croatian National Theatre. Flag Coat of arms Zagreb Location of Zagreb in Croatia Show map of Croatia Zagreb Zagreb (Europe) Show map of Europe Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Coordinates: 45°49′N 15°59′E  /  45.817°N 15.983°E  / 45.817; 15.983 Country   Croatia County City of Zagreb RC diocese 1094 Free royal city 1242 Unified 1850 Subdivisions 17 city districts 218 local committees (70 settlements) Government  • Type Mayor-Council  • Mayor Milan Bandić (BM 365)  • City Assembly 51 members BM 365, ZL, NS-R (14) SDP, GLAS, HNS, HSS, NH-PS (13) ...
Read more

Understanding the information contained in the Deep Space Network XML data?

Image
up vote 1 down vote favorite NASA's Deep Space Network activity can be viewed conveniently at https://eyes.nasa.gov/dsn/dsn.html but if you would like data in a more raw XML form, it can be accessed as described in this answer Below is a small Python script that reads the XML into a list of dictionaries, and saves it to disk as a JSON. The dictionaries are organized at the higher level by dish, then by spacecraft per dish. Question: What are the most-likely meanings of all the data field contained in the Deep Space Network XML data? This is the example url from the other answer, used in the script below: https://eyes.nasa.gov/dsn/data/1365107113.xml def dictify(r,root=True): # https://stackoverflow.com/questions/2148119/how-to-convert-an-xml-string-to-a-dictionary-in-python/30923963#30923963 if root: ret...
Read more