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

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 ${{\text {P}}^m}$.