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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Frobenius calculations of Picard groups and the Birch-Tate-Swinnerton-Dyer conjecture


Author: Raymond T. Hoobler
Journal: Trans. Amer. Math. Soc. 222 (1976), 345-352
MSC: Primary 14C20; Secondary 14G20
MathSciNet review: 0437529
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Abstract: Let $ Y \subset {{\text{P}}^m}$ be a subvariety of codimension d defined by an ideal I in char $ p > 0$ with $ {H^1}(Y,\mathcal{O}( - 1)) = 0$. If t is an integer greater than $ {\log _p}(d)$ and $ {H^i}(Y,{I^n}/{I^{n + 1}}) = 0$ for $ n > > 0$ and $ i = 1,2$, then $ {\text{Pic}}(Y)$ is an extension of a finite p-primary group of exponent at most $ {p^t}$ by $ Z[\mathcal{O}(1)]$ and $ {\text{Br}}'(Y)(p)$ is a group of exponent at most $ {p^t}$. If Y is also smooth and defined over a finite field with $ \dim Y < p$ and $ p \ne 2$, then the B-T-SD conjecture holds for cycles of codimension 1. These results are proved by studying the etale cohomology of the Frobenius neighborhoods of Y in $ {{\text{P}}^m}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0437529-7
PII: S 0002-9947(1976)0437529-7
Keywords: Frobenius neighborhood, Brauer group
Article copyright: © Copyright 1976 American Mathematical Society