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 κ(C) ∈ CH₁(J)⊗ Q in the Chow group of 1-cycles on J. Namely, $(A)$ If Hₚᵣᵢₘ³(J)Aut(C) = 0, then κ(C) vanishes; $(B)$ If H⁰(J, ΩJ³)Aut(C) = 0 and the Hodge conjecture holds, then κ(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.