  Let $C$ be a smooth projective curve, and let $J$ be its Jacobian. We prove
vanishing criteria for the Ceresa cycle $\kappa(C) \in \mathrm{CH}_1(J)\otimes
\mathbb{Q}$ in the Chow group of 1-cycles on $J$. Namely,
  $(A)$ If $\mathrm{H}_{\mathrm{prim}}^3(J)^{\mathrm{Aut}(C)} = 0$, then
$\kappa(C)$ vanishes;
  $(B)$ If $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$ and the Hodge
conjecture holds, then $\kappa(C)$ vanishes modulo algebraic equivalence.
  We then study the first interesting case where $(B)$ holds but $(A)$ does
not, namely the case of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$.
Using work of Schoen on the Hodge conjecture, we show that the Ceresa cycle of
a Picard curve is torsion in the Griffiths group. Moreover, we determine
exactly when it is torsion in the Chow group. As a byproduct, we show that
there are infinitely many plane quartic curves over $\mathbb{Q}$ with torsion
Ceresa cycle (in fact, there is a one parameter family of such curves).
Finally, we determine which automorphism group strata are contained in the
vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$.
