2007/01/22 by Mirela Çiperiani, Ciperiani, Mirela, Daniel Krashen +1 · 1 citation
Arts and Humanities · Mathematics · Social Sciences · #11G05 #16K20 #16K50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical and Political Studies #North African History and Literature #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.math/0701614
openalex publication_date 2007/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop techniques for computing the relative Brauer group of curves, focusing particularly on the case where the genus is 1. We use these techniques to show that the relative Brauer group may be infinite (for certain ground fields) as well as to determine this group explicitly for certain curves defined over the rational numbers. To connect to previous descriptions of relative Brauer groups in the literature, we describe a family of genus 1 curves, which we call "cyclic type" for which the relative Brauer group can be shown to have a particularly nice description. In order to do this, we discuss a number of formulations of the pairing between the points on an elliptic curve and its Weil-Chatêlet group into the Brauer group of the ground field, and draw connections to the period-index problem for genus 1 curves.