Non-vanishing of Ceresa and Gross--Kudla--Schoen cycles associated to modular curves

Given an algebraic curve X of genus ≥ 3, one can construct two algebraic $1$-cycles, the Ceresa cycle and the Gross-Kudla-Schoen modified diagonal cycle, each living in the Jacobian of X and the triple product X × X × X respectively. These two cycles are homologically trivial but are of infinite order in their corresponding Chow groups for a very general curve over C. From the work of S-W Zhang, for a fixed curve these two cycles are non-torsion in their corresponding Chow groups if and only if one of them is. In this paper, we prove that the Ceresa and Gross-Kudla-Schoen cycles associated to a modular curve X are non-torsion in the corresponding Chow groups when X=Γ H̄ for certain congruence subgroups Γ⊂ SL₂(Z). We obtain the result by studying a pullback formula for special divisors by the diagonal map X X × X.