Distribution of torsion subgroups of elliptic curve
$begingroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
$endgroup$
add a comment |
$begingroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
$endgroup$
add a comment |
$begingroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
$endgroup$
By Mazur's theorem, we know the list of possible torsion subgroups of elliptic curves over $mathbb{Q}$. Now, if we order them by height, can we compute the distribution of each possible groups? According to wikipedia, it is known that each group occurs infinitely many times, so maybe all of the possible groups have positive density.
elliptic-curves
elliptic-curves
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
asked Dec 5 '18 at 23:38
Seewoo LeeSeewoo Lee
6,247826
6,247826
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
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: "69"
};
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027832%2fdistribution-of-torsion-subgroups-of-elliptic-curve%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
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
$endgroup$
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
$endgroup$
Curves with no nontrivial torsion are density 1 in the set of all elliptic curves over $mathbf{Q}$. This is a result of Duke from 1997 (paper is titled Courbes elliptiques sur Q sans nombres premiers exceptionnels).
More specifically, by work of Harron and Snowden, we actually know the leading term in the asymptotic growth of the number of elliptic curves with a given torsion subgroup ordered by height. See Theorems 1.1 and 1.5 in their article.
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
answered Dec 16 '18 at 4:39
Brandon CarterBrandon Carter
7,23322538
7,23322538
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
$begingroup$
This is exactly I wanted!
$endgroup$
– Seewoo Lee
Dec 17 '18 at 4:04
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- 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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3027832%2fdistribution-of-torsion-subgroups-of-elliptic-curve%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
