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 be a subvariety of codimension *d* defined by an ideal *I* in char with . If *t* is an integer greater than and for and , then is an extension of a finite *p*-primary group of exponent at most by and is a group of exponent at most . If *Y* is also smooth and defined over a finite field with and , 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 .

**[1]**Spencer Bloch,*Algebraic 𝐾-theory and crystalline cohomology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 187–268 (1978). MR**488288****[2]**Robin Hartshorne,*Ample subvarieties of algebraic varieties*, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR**0282977****[3]**Raymond T. Hoobler,*Cohomology of purely inseparable Galois coverings*, J. Reine Angew. Math.**266**(1974), 183–199. MR**0364258****[4]**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**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1976-0437529-7

Keywords:
Frobenius neighborhood,
Brauer group

Article copyright:
© Copyright 1976
American Mathematical Society