  A smooth plane curve is said to admit a symmetric determinantal
representation if it can be defined by the determinant of a symmetric matrix
with entries in linear forms in three variables. We study the local-global
principle for the existence of symmetric determinantal representations of
smooth plane curves over a global field of characteristic different from two.
When the degree of the plane curve is less than or equal to three, we relate
the problem of finding symmetric determinantal representations to more familiar
Diophantine problems on the Severi-Brauer varieties and mod 2 Galois
representations, and prove that the local-global principle holds for conics and
cubics. We also construct counterexamples to the local-global principle for
quartics using the results of Mumford, Harris, and Shioda on theta
characteristics.
