An abelian variety not isogenous to a Jacobian
8
$begingroup$
In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $mathbb F_5$.
The two ways they suggest this is usually checked is:
- Show that a point count of the associated virtual curve of the abelian variety is negative.
- Show that there is an extension $mathbb F_{p^d}subset mathbb F_{p^n}$ such that the curve has fewer points over the bigger field than over the smaller field.
Neither of these hold for the example linked above. I also checked that the Weil bound for a genus $2$ curve over $mathbb F_{p^d}$ holds for the first few virtual point counts.
- How are they concluding that this isogeny class doesn't contain a Jacobian?
- More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?
algebraic-geometry algebraic-number-theory finite-fields arithmetic-geometry abelian-varieties
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
$endgroup$
add a comment |
8
$begingroup$
In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $mathbb F_5$.
The two ways they suggest this is usually checked is:
- Show that a point count of the associated virtual curve of the abelian variety is negative.
- Show that there is an extension $mathbb F_{p^d}subset mathbb F_{p^n}$ such that the curve has fewer points over the bigger field than over the smaller field.
Neither of these hold for the example linked above. I also checked that the Weil bound for a genus $2$ curve over $mathbb F_{p^d}$ holds for the first few virtual point counts.
- How are they concluding that this isogeny class doesn't contain a Jacobian?
- More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?
algebraic-geometry algebraic-number-theory finite-fields arithmetic-geometry abelian-varieties
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
$endgroup$
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
1
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16
add a comment |
8
8
8
1
$begingroup$
In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $mathbb F_5$.
The two ways they suggest this is usually checked is:
- Show that a point count of the associated virtual curve of the abelian variety is negative.
- Show that there is an extension $mathbb F_{p^d}subset mathbb F_{p^n}$ such that the curve has fewer points over the bigger field than over the smaller field.
Neither of these hold for the example linked above. I also checked that the Weil bound for a genus $2$ curve over $mathbb F_{p^d}$ holds for the first few virtual point counts.
- How are they concluding that this isogeny class doesn't contain a Jacobian?
- More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?
algebraic-geometry algebraic-number-theory finite-fields arithmetic-geometry abelian-varieties
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
$endgroup$
In the L-functions and Modular Forms Database is an isogeny class of an abelian variety of dimension $2$ over $mathbb F_5$. They claim that it is principally polarizable but not the Jacobian of a curve over $mathbb F_5$.
The two ways they suggest this is usually checked is:
- Show that a point count of the associated virtual curve of the abelian variety is negative.
- Show that there is an extension $mathbb F_{p^d}subset mathbb F_{p^n}$ such that the curve has fewer points over the bigger field than over the smaller field.
Neither of these hold for the example linked above. I also checked that the Weil bound for a genus $2$ curve over $mathbb F_{p^d}$ holds for the first few virtual point counts.
- How are they concluding that this isogeny class doesn't contain a Jacobian?
- More generally, what other techniques are there that help us rule out a principally polarized abelian variety being isogenous to a Jacobian?
algebraic-geometry algebraic-number-theory finite-fields arithmetic-geometry abelian-varieties
algebraic-geometry algebraic-number-theory finite-fields arithmetic-geometry abelian-varieties
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
asked Dec 6 '18 at 17:34
RaviRavi
7,2751738
7,2751738
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
1
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16
add a comment |
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
1
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
1
1
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16
add a comment |
0
active
oldest
votes
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
});
}
});
draft saved
draft discarded
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%2f3028802%2fan-abelian-variety-not-isogenous-to-a-jacobian%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3028802%2fan-abelian-variety-not-isogenous-to-a-jacobian%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
$begingroup$
In principle, at least, one can enumerate all curves of a given genus over $mathbb{F}_5$, and compute their Frobenius polynomials. If none of them match the given curve, then you're done.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:21
$begingroup$
PS: sorry, I meant "if none of them match the given abelian variety", of course.
$endgroup$
– David Loeffler
Dec 8 '18 at 15:39
$begingroup$
@DavidLoeffler Hi David. Thank you. Your method works but I was hoping it would be some invariant of the AV that would be less brute force, something in the vein of a virtual point count.
$endgroup$
– Ravi
Dec 9 '18 at 17:42
1
$begingroup$
I agree, I'd also be happy to see a more conceptual approach.
$endgroup$
– David Loeffler
Dec 9 '18 at 18:16