Number of solutions mod p and Betti numbersBetti numbers from virtual Betti numbers of a cell decompositionComparison between singular and etale cohomology in Batyrev's paper on Birational Calabi-Yau varietiesHodge numbers of reduction mod $p$Betti numbers of Proper nonprojective varietiesUpper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?Relatively concise English expositions of the proofs of the various Weil conjecturesBounds on Betti numbers of subvarieties?Weight of Weil numbers in the residue field of $overlinemathbbQ_l$Current status of independence of Betti numbers for different Weil cohomology theories
Number of solutions mod p and Betti numbers
Betti numbers from virtual Betti numbers of a cell decompositionComparison between singular and etale cohomology in Batyrev's paper on Birational Calabi-Yau varietiesHodge numbers of reduction mod $p$Betti numbers of Proper nonprojective varietiesUpper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)Why is there a Parity Problem in Sieve Theory and not a Mod p problem for any other p?Relatively concise English expositions of the proofs of the various Weil conjecturesBounds on Betti numbers of subvarieties?Weight of Weil numbers in the residue field of $overlinemathbbQ_l$Current status of independence of Betti numbers for different Weil cohomology theories
$begingroup$
Suppose $X$ is proper flat scheme over Sepc$mathbbZ$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(mathbbF_p^n)$ for all $n$. My question is, what if instead of fix a prime and vary $n$, we fix an $n$ and vary $p$?
More precisely, let $N$ be a fixed number, suppose we know the size of $X(mathbbF_p^n)$ for all $n$ between 1 and $N$, and (almost) all prime $p$. Can we recover the Betti numbers of $X$?
ag.algebraic-geometry nt.number-theory
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Suppose $X$ is proper flat scheme over Sepc$mathbbZ$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(mathbbF_p^n)$ for all $n$. My question is, what if instead of fix a prime and vary $n$, we fix an $n$ and vary $p$?
More precisely, let $N$ be a fixed number, suppose we know the size of $X(mathbbF_p^n)$ for all $n$ between 1 and $N$, and (almost) all prime $p$. Can we recover the Betti numbers of $X$?
ag.algebraic-geometry nt.number-theory
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Suppose $X$ is proper flat scheme over Sepc$mathbbZ$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(mathbbF_p^n)$ for all $n$. My question is, what if instead of fix a prime and vary $n$, we fix an $n$ and vary $p$?
More precisely, let $N$ be a fixed number, suppose we know the size of $X(mathbbF_p^n)$ for all $n$ between 1 and $N$, and (almost) all prime $p$. Can we recover the Betti numbers of $X$?
ag.algebraic-geometry nt.number-theory
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Suppose $X$ is proper flat scheme over Sepc$mathbbZ$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(mathbbF_p^n)$ for all $n$. My question is, what if instead of fix a prime and vary $n$, we fix an $n$ and vary $p$?
More precisely, let $N$ be a fixed number, suppose we know the size of $X(mathbbF_p^n)$ for all $n$ between 1 and $N$, and (almost) all prime $p$. Can we recover the Betti numbers of $X$?
ag.algebraic-geometry nt.number-theory
ag.algebraic-geometry nt.number-theory
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
HeavensfallHeavensfall
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
HeavensfallHeavensfall
311
asked 8 hours ago
HeavensfallHeavensfall
311
311
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Heavensfall is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For the first claim, I think you want to also assume $X$ is smooth (away from some finite set of primes, not including $p$).
For the second claim, again with the assumption that $X$ is smooth, this would follow from a weak form of the generalized Sato-Tate conjecture. But nothing like this is known currently without very strong additional assumptions on $X$.
See Nick Katz's article Simple Things We Don't Know which is about almost exactly this question.
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
$endgroup$
add a comment |
Your Answer
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
);
);
Heavensfall is a new contributor. Be nice, and check out our Code of Conduct.
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%2fmathoverflow.net%2fquestions%2f334289%2fnumber-of-solutions-mod-p-and-betti-numbers%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
$begingroup$
For the first claim, I think you want to also assume $X$ is smooth (away from some finite set of primes, not including $p$).
For the second claim, again with the assumption that $X$ is smooth, this would follow from a weak form of the generalized Sato-Tate conjecture. But nothing like this is known currently without very strong additional assumptions on $X$.
See Nick Katz's article Simple Things We Don't Know which is about almost exactly this question.
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
$endgroup$
add a comment |
$begingroup$
For the first claim, I think you want to also assume $X$ is smooth (away from some finite set of primes, not including $p$).
For the second claim, again with the assumption that $X$ is smooth, this would follow from a weak form of the generalized Sato-Tate conjecture. But nothing like this is known currently without very strong additional assumptions on $X$.
See Nick Katz's article Simple Things We Don't Know which is about almost exactly this question.
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
$endgroup$
add a comment |
$begingroup$
For the first claim, I think you want to also assume $X$ is smooth (away from some finite set of primes, not including $p$).
For the second claim, again with the assumption that $X$ is smooth, this would follow from a weak form of the generalized Sato-Tate conjecture. But nothing like this is known currently without very strong additional assumptions on $X$.
See Nick Katz's article Simple Things We Don't Know which is about almost exactly this question.
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
$endgroup$
For the first claim, I think you want to also assume $X$ is smooth (away from some finite set of primes, not including $p$).
For the second claim, again with the assumption that $X$ is smooth, this would follow from a weak form of the generalized Sato-Tate conjecture. But nothing like this is known currently without very strong additional assumptions on $X$.
See Nick Katz's article Simple Things We Don't Know which is about almost exactly this question.
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
answered 7 hours ago
Will SawinWill Sawin
70.3k7142291
70.3k7142291
add a comment |
add a comment |
Heavensfall is a new contributor. Be nice, and check out our Code of Conduct.
Heavensfall is a new contributor. Be nice, and check out our Code of Conduct.
Heavensfall is a new contributor. Be nice, and check out our Code of Conduct.
Heavensfall is a new contributor. Be nice, and check out our Code of Conduct.
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.
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%2fmathoverflow.net%2fquestions%2f334289%2fnumber-of-solutions-mod-p-and-betti-numbers%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
