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COMPARING THE BRAUER GROUP TO THE TATE–SHAFAREVICH GROUP

Part of: Arithmetic problems. Diophantine geometry Surfaces and higher-dimensional varieties Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 July 2018

Thomas H. Geisser Open the ORCID record for Thomas H. Geisser [Opens in a new window]
Thomas H. Geisser*
Affiliation:
Rikkyo University, Ikebukuro, Tokyo, Japan (geisser@rikkyo.ac.jp)
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Abstract

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

Keywords

Brauer groupTate–Shafarevich group

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G17: Positive characteristic ground fields 14J20: Arithmetic ground fields 11G25: Varieties over finite and local fields 11G35: Varieties over global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 965 - 970
Copyright
© Cambridge University Press 2018

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