Groups of components of Néron models of Jacobians and Brauer groups
Saikat Biswas
TL;DR
The paper addresses how the component group $Φ_A$ of the Néron model of the Jacobian $A$ of a curve $X$ over a non-archimedean local field $K$ is connected to the Brauer group $\,Br(X)$. It proves an exact sequence $0 \to \mathrm{Hom}(Br_{nr}(X)/Br_0(X), Q/Z) \to Φ_A(k) \to Z/dZ \to 0$ with $d=δ'/δ^{nr}'$, linking local Brauer invariants to the Tamagawa component group. As a consequence, $Br_{nr}(X)/Br_0(X)$ is finite of order $c_A/d$, and the paper discusses global-field corollaries that relate Shafarevich–Tate groups, Tamagawa numbers, and Brauer data in a Birch–Swinnerton-Dyer–type framework. The approach combines Picard group sequences, Néron model theory, and dualities, establishing a local-global bridge between Brauer groups and Tamagawa components with potential arithmetic applications.
Abstract
Let $X$ be a proper, smooth, and geometrically connected curve over a non-archimedean local field $K$. In this paper, we relate the component group of the Néron model of the Jacobian of $X$ to the Brauer group of $X$.