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Abstract LetL/Kbe a Galois extension of number fields with Galois groupG. We show that if the density of prime ideals inKthat split totally inLtends to 1/∣G∣ with a power saving error term, then the density of prime ideals inKwhose Frobenius is a given conjugacy classC⊂Gtends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros ofζL(s)/ζK(s).
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The Cohomological Brauer Group of a Torsion ${\mathbb{G}}_{m}$-Gerbe
Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.
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- Award ID(s):
- 1646385
- PAR ID:
- 10123040
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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