Comparing the Brauer group and the Tate Shafarevich group

2017/12/18 by Geisser, Thomas
#11G25 #11G35 #11G40 #14G17 #14J20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.06249

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.

Related

• Tate's conjecture and the Tate-Shafarevich group over global function fields
• Brauer groups and Tate-Shafarevich groups
• Brauer and Neron-Severi groups of surfaces over finite fields
• The Brauer Group of a Surface over a Finite Field
• On geometric Brauer groups and Tate-Shafarevich groups