Vanishing criteria for Ceresa cycles

Let C be a smooth projective curve, and let J be its Jacobian. We prove vanishing criteria for the Ceresa cycle (C) CH₁ (J) Q in the Chow group of 1-cycles on J. Namely, (A) If H₏ₑ₈₌³ (J) ^Aut (C) = 0, then (C) vanishes; (B) If H⁰ (J, J³) ^Aut (C) = 0 and the Hodge conjecture holds, then (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 y³ = x⁴ + ax² + 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 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 M₃.