  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-Chat\^elet group into the Brauer group of the ground field, and
draw connections to the period-index problem for genus 1 curves.
