  The purpose of this article is to study the distribution of the trace on the
unitary symplectic group. We recall its relevance to equidistribution results
for the eigenvalues of the Frobenius in families of abelian varieties over
finite fields, and to the limiting distribution of the number of points of
curves. We give four expressions of the trace distribution if g = 2, in terms
of special functions, and also an expression of the distribution of the trace
in terms of elementary symmetric functions. In an appendix, we prove a formula
for the trace of the exterior power of the identity representation.
