Timeline for Arithmetic zeta function and local zeta functions
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 3, 2017 at 11:52 | comment | added | reuns | When you replace $\mathbb{Z},\mathbb{Q}$ by $\mathbb{Z}[i],\mathbb{Q}(i)$ you are also replacing $\zeta_{X |\mathbb{F}_p}(s)$ by $\zeta_{X |\mathbb{F}_p^2}(s)$ for $p \equiv 3 \bmod 4$ and by $ \zeta_{X |\mathbb{F}_p}(s)^2$ for $p \equiv 1 \bmod 4$. | |
| Nov 20, 2015 at 11:22 | history | edited | user9072 | CC BY-SA 3.0 |
deleted 8 characters in body; edited tags
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| Nov 20, 2015 at 9:06 | comment | added | THC | I am sorry for the confusion - it is enough to suppose that $X$ is a scheme of finite type over $\mathrm{Spec}(\mathbb{Z})$. | |
| Nov 20, 2015 at 1:00 | comment | added | R. van Dobben de Bruyn | I am fine with the arithmetic zeta function, but I was confused by the term nonsingular variety. For me, varieties are over a field, in which case there is at most one prime. But I guess the obvious modification is that $X$ is some scheme over $\operatorname{Spec} \mathbb Z$ whose generic fibre is smooth (i.e. nonsingular) (rather than the modification I first considered where $X \to \operatorname{Spec} \mathbb Z$ is smooth). | |
| Nov 19, 2015 at 21:27 | comment | added | Daniel Loughran | @R. van Dobben de Bruyn: This is the standard definition of an arithmetic zeta function: en.wikipedia.org/wiki/Arithmetic_zeta_function. No good reduction hypotheses are required. The OP should really however be choosing a model for $X$ over $\mathbb{Z}$. | |
| Nov 19, 2015 at 16:26 | comment | added | R. van Dobben de Bruyn | I don't understand why you are taking the product over all primes. Do you mean that $X$ is a smooth projective scheme over $\operatorname{Spec} \mathbb Z$? (that seems very strong; I don't know many examples that have good reduction at all primes...) | |
| Nov 19, 2015 at 16:09 | history | asked | THC | CC BY-SA 3.0 |