|
8
|
|
edited May 8 2011 at 14:25 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map
$$H^{2d}c(X,\mathbf{Q}{\ell})\to\mathbf{Q}_{\ell}(-d),$$
$
H^{2d}_c(X,\mathbf{Q}_{\ell})\to\mathbf{Q}_{\ell}(-d),
$$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
7
|
|
edited Dec 15 2010 at 12:09 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map $H^{2d}$H^{2d}c(X,\mathbf{Q}\ell)\to\mathbf{Q}_\ell(-d),$
{\ell})\to\mathbf{Q}_{\ell}(-d),$$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
6
|
|
edited Dec 15 2010 at 11:46 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map H^{2d}_c(X, Q_l)\to Q_l(-d),
$H^{2d}c(X,\mathbf{Q}\ell)\to\mathbf{Q}_\ell(-d),$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
5
|
|
edited Dec 13 2010 at 18:15 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in \$\pi_1(X)$, $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map $H^{2d}_c(X,\mathbb Q_{\ell})\to\mathbb Q_{\ell}(-d)$H^{2d}_c(X, Q_l)\to Q_l(-d),
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
4
|
|
edited Dec 13 2010 at 18:14 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, \$\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map \$H^{2d}c(X,\mathbb Q{\ell})\to\mathbb Q_l(-d),\$
$H^{2d}_c(X,\mathbb Q_{\ell})\to\mathbb Q_{\ell}(-d)$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
3
|
|
edited Dec 13 2010 at 18:12 |
For a variety $X$ over a finite field, I guess one can take $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point $x$ (like integral over a little loop around that point) is the trace of the local Frobenius $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that $Frob_x$ can be regarded as an element (or conjugacy class) in $\pi_1(X)$, "a loop around $x$". The global integral would be the global trace map $H^{2d}\$H^{2d}c(X,\mathbb Q{\ell})\to\mathbb Q_{\ell}(-d),$
Q_l(-d),\$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point $v$ in $Spec\ O_k$ still defines a "loop" $Frob_v$ in $\pi_1(Spec \pi_1(Spec\ k)$ (let's allow ramified covers. One can take the image of $Frob_v$ under $\pi_1(Spec \pi_1(Spec\ k)\to\pi_1(Spec\ O_k)$, but the target group doesn't seem to be big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec O_k,G_m)\to\mathbb{Q/Z}$ H^3(Spec\ O_k,\mathbb G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
2
|
|
edited Dec 13 2010 at 18:05 |
For a variety X $X$ over a finite field, I guess one can take \ell-adic $\ell$-adic sheaves to replace differential forms. Then the local integral around a closed point x $x$ (like integral over a little loop around that point) is the trace of the local Frobenius Frob_x $Frob_x$ on the stalk of sheaf, the so-called naive local term. Note that Frob_x $Frob_x$ can be regarded as an element (or conjugacy class) in \pi_1(X), $\pi_1(X)$, "a loop around x". $x$". The global integral would be the global trace map H^{2d}c(X,Q\ell) --> Q_\ell (-d), $H^{2d}c(X,\mathbb Q{\ell})\to\mathbb Q_{\ell}(-d),$
and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the (2\pi i)^d $(2\pi i)^d$ one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point v $v$ in Spec O_k $Spec\ O_k$ still defines a "loop" Frob_v $Frob_v$ in \pi_1(Spec k) $\pi_1(Spec k)$ (let's allow ramified covers. One can take the image of Frob_v $Frob_v$ under \pi_1(Spec k) --> \pi_1(Spec O_k)$\pi_1(Spec k)\to\pi_1(Spec O_k)$, but the target group doesn't seem to be big)big enough). For global integral, there's the Artin-Verdier trace map $H^3(Spec O_k,G_m) --> Q/Z O_k,G_m)\to\mathbb{Q/Z}$ and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
|
|
1
|
|
answered Nov 4 2009 at 19:50 |
For a variety X over a finite field, I guess one can take \ell-adic sheaves to replace differential forms. Then the local integral around a closed point x (like integral over a little loop around that point) is the trace of the local Frobenius Frob_x on the stalk of sheaf, the so-called naive local term. Note that Frob_x can be regarded as an element (or conjugacy class) in \pi_1(X), "a loop around x". The global integral would be the global trace map H^{2d}c(X,Q\ell) --> Q_\ell (-d), and the Tate twist is responsible for the Hodge structure in Betti cohomology (or the (2\pi i)^d one has to divide by). The Lefschetz trace formula might be the analog of the residue theorem in complex analysis on Riemann surfaces.
For the case of number fields, each closed point v in Spec O_k still defines a "loop" Frob_v in \pi_1(Spec k) (let's allow ramified covers. One can take the image of Frob_v under \pi_1(Spec k) --> \pi_1(Spec O_k), but the target group doesn't seem to be big). For global integral, there's the Artin-Verdier trace map H^3(Spec O_k,G_m) --> Q/Z and a "Poincar\'e duality" in this setting, but I don't know if there is a trace formula. The fact that 3 is odd always makes me excited and confused.
So basically I think of trace maps (both local and global) as counterpart of integrals. Correct me if I was wrong.
|
|
|
