  Given a field with a set of discrete valuations $V$, we show how the genus of
a division algebra over the field is related to the genus of the residue
algebras at various valuations in $V$ and the ramification data. When the
division algebra is a quaternion, we show the triviality of genus over many
fields which include higher local fields, function fields of curves over higher
local fields and function fields of curves over real closed fields. We also
consider function fields of curves over global fields with a rational point and
show how the genus problem is related to the $2$-torsion of the
Tate-Shafarevich group of its Jacobian. As a special case, we show how the
methods developed yield better bounds on the size of the genus over function
fields of elliptic curves and demonstrate how they can be computed directly
using arithmetic data of the elliptic curve with a number of examples.
