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

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- by Raymond T. Hoobler PDF
- Trans. Amer. Math. Soc.
**222**(1976), 345-352 Request permission

## 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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*Cohomology of purely inseparable Galois coverings*, J. Reine Angew. Math.**266**(1974), 183–199. MR**364258**, DOI 10.1515/crll.1974.266.183 - John T. Tate,
*Algebraic cycles and poles of zeta functions*, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 93–110. MR**0225778**

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 345-352 - MSC: Primary 14C20; Secondary 14G20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0437529-7
- MathSciNet review: 0437529