Frobenius Calculations of Picard Groups and the Birch - Tate - Swinnerton Dyer Conjecture

Let Y C Pm be a subvariety of codimension d defined by an ideal / in charp > 0 with //'(Y, 0 (-1)) = 0. If t is an integer greater than log (d) and Hi( Y, f/l"+i) = 0 for n » 0 and i =1,2, then Pic(Y) is an extension of a finite p-primary group of exponent at most pt by Z[ 0 (1 )1 and Br'(Y)(p) is a group of exponent at most pl'. If Y is also smooth and defined over a finite field with dim Y < p and p i= 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 Pm. Let i : Y —»• ?m be a smooth subvariety of Pm over C. If dim(JO > Vi(m + 2), then Barth and Larsen have shown that Vic(Y) = Z and it is generated by Öy-(l). This follows easily from their isomorphism theorems for the cohomology of smooth subvarieties of low codimension and the exponential sequence. Ogus has given an algebraic proof of this result in characteristic 0 by studying the cohomology of the formal completion V of Pm along Y and using the exponential map to pass from Pic^) to Pic(Y). We use the same approach via V, but the techniques required to study Pic(Y) are quite different in char p > 0. Clearly a general exponential map does not exist. Instead we have the Frobenius map. Moreover if Y0 C Y is a square zero deformation, then the truncated exponential sequence shows that Pic(Y) —► Pic(Y0) has a p-torsion kernel and cokernel. In particular if i : Y —► Pm is a closed embedding defined by an ideal / and V is the formal scheme obtained by completing Pm along Y, then Pici^) has the same rank for any N but Pic(y) may have smaller rank. This is essentially due to a p-adic limit phenomenon wherein the limit may be zero without the groups being 0. By requiring that H'(Y, In/In+1) = 0 for 1 = 1,2 and H\Y, 0Y(-1)) = 0, we can avoid this and can obtain a remarkably strong theorem-Pic(Y) has rank 1 and contains no p" torsion for n > log (d) where d = codim(y, Pm). Moreover we verify the BirchTate-Swinnerton Dyer conjectures in codimension 1 for such smooth varieties Received by the editors March 31, 1975. AMS (MOS) subject classifications (1970). Primary 14C20, 14Fxx.