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
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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.
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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 |
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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 6 hours ago
Will SawinWill Sawin
70.3k7142291
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1 Answer
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$begingroup$
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 6 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 6 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 6 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 6 hours ago
Will SawinWill Sawin
70.3k7142291
answered 6 hours ago
Will SawinWill Sawin
70.3k7142291
answered 6 hours ago
Will SawinWill Sawin
70.3k7142291
answered 6 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.
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