Corrigendum to Néron models, Lie algebras, and reduction of curves of genus one [LLR1] and The Brauer group of a surface [LLR2]

Abstract

Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part on a result of Gordon [Gor], which we used in [LLR1] to establish a key formula. Thomas Geisser noted that the formula in [LLR1] is incorrect, due to an omission in [Gor], and provides a corrected formula. We explain in this corrigendum how to modify the work of Gordon to establish a correct formula. The corrected formula can be used to prove the result in [LLR2] without further modifications.

To appear in Invent. Math. In the new introduction, we make clear that Thomas Geisser in [4] already obtained the correct formula. His method also applies to the number field case (up to a power of 2 if not totally real)