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Ceresa Cycles of X₀(N)
Elvira Lupoian, James Rawson
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The Ceresa cycle is an algebraic 1-cycle on the Jacobian of an algebraic curve. Although it is homologically trivial, Ceresa famously proved that for a very general complex curve of genus at least 3, it is non-trivial in the Chow group. In this paper we study the Ceresa cycle attached to the complete modular curve X₀(N) modulo rational equivalence. For prime level p we give a complete description, namely we prove that if X₀(p) is not hyperelliptic, then its Ceresa cycle is non-torsion. For general level N, we prove that there are finitely many X₀(N) with torsion Ceresa cycle. Our method relies on the relationship between the vanishing of the Ceresa cycle and Chow-Heegner points on the Jacobian. We use the geometry and arithmetic of modular Jacobians to prove that such points are of infinite order and therefore deduce non-vanishing of the Ceresa cycle.
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