How to compute the $p$-adic period of $mathbb G_m$?
$begingroup$
I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $mathbb G_m$:
I know the example for the infinite place: let $w=frac{dz}{z}$ be the specific holomorphic differential on $mathbb G_m(mathbb C)=mathbb C^{times}$, $S^1$ be the unit circle in $mathbb C^{times}$ which is a generator for $H_1(mathbb C^{times}, mathbb Z)$, then the period is the intergal:
$int_{S^1} w= int_{S^1} frac{dz}{z}=2 pi i.$
which is clear from standard complex analysis. My question is how to compute the $p$-adic period as the comparison theorem in $p$-adic Hodge theory he used:
I think this is a standard exercise in $p$-adic Hodge theory, but I couldn't find some references about the computation $|t_p|_p=p^{-frac{1}{p-1}}$ (here the p-adic valuation is normalized such that $|p|_p=1/p$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.
Thank you for any help.
number-theory p-adic-number-theory arithmetic-geometry galois-representations
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
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add a comment |
$begingroup$
I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $mathbb G_m$:
I know the example for the infinite place: let $w=frac{dz}{z}$ be the specific holomorphic differential on $mathbb G_m(mathbb C)=mathbb C^{times}$, $S^1$ be the unit circle in $mathbb C^{times}$ which is a generator for $H_1(mathbb C^{times}, mathbb Z)$, then the period is the intergal:
$int_{S^1} w= int_{S^1} frac{dz}{z}=2 pi i.$
which is clear from standard complex analysis. My question is how to compute the $p$-adic period as the comparison theorem in $p$-adic Hodge theory he used:
I think this is a standard exercise in $p$-adic Hodge theory, but I couldn't find some references about the computation $|t_p|_p=p^{-frac{1}{p-1}}$ (here the p-adic valuation is normalized such that $|p|_p=1/p$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.
Thank you for any help.
number-theory p-adic-number-theory arithmetic-geometry galois-representations
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
$endgroup$
add a comment |
$begingroup$
I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $mathbb G_m$:
I know the example for the infinite place: let $w=frac{dz}{z}$ be the specific holomorphic differential on $mathbb G_m(mathbb C)=mathbb C^{times}$, $S^1$ be the unit circle in $mathbb C^{times}$ which is a generator for $H_1(mathbb C^{times}, mathbb Z)$, then the period is the intergal:
$int_{S^1} w= int_{S^1} frac{dz}{z}=2 pi i.$
which is clear from standard complex analysis. My question is how to compute the $p$-adic period as the comparison theorem in $p$-adic Hodge theory he used:
I think this is a standard exercise in $p$-adic Hodge theory, but I couldn't find some references about the computation $|t_p|_p=p^{-frac{1}{p-1}}$ (here the p-adic valuation is normalized such that $|p|_p=1/p$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.
Thank you for any help.
number-theory p-adic-number-theory arithmetic-geometry galois-representations
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
$endgroup$
I am reading one paper by Colmez about periods of abelian varieties with complex multiplication. As a motivation, he says we can compute the periods of $mathbb G_m$:
I know the example for the infinite place: let $w=frac{dz}{z}$ be the specific holomorphic differential on $mathbb G_m(mathbb C)=mathbb C^{times}$, $S^1$ be the unit circle in $mathbb C^{times}$ which is a generator for $H_1(mathbb C^{times}, mathbb Z)$, then the period is the intergal:
$int_{S^1} w= int_{S^1} frac{dz}{z}=2 pi i.$
which is clear from standard complex analysis. My question is how to compute the $p$-adic period as the comparison theorem in $p$-adic Hodge theory he used:
I think this is a standard exercise in $p$-adic Hodge theory, but I couldn't find some references about the computation $|t_p|_p=p^{-frac{1}{p-1}}$ (here the p-adic valuation is normalized such that $|p|_p=1/p$). By the way, the original paper is Périodes des variétés abéliennes à multiplication complexe, Ann. of Maths 138 (1993), 625–683, which can be found on Colmez's website.
Thank you for any help.
number-theory p-adic-number-theory arithmetic-geometry galois-representations
number-theory p-adic-number-theory arithmetic-geometry galois-representations
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
edited Dec 14 '18 at 0:21
zzy
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
asked Dec 13 '18 at 21:49
zzyzzy
2,6021420
2,6021420
add a comment |
add a comment |
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