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Testing the non-vanishing of the Ceresa cycle of X0(N)X_0(N)

Primarily human-written textHuman understanding: all partsmath.AG — Algebraic Geometrymath.NT — Number Theory

Contributed by Leonardo Finzi ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

For a curve CC and a fixed degree-one divisor ee of CC, we can embed CC into its Jacobian. This embedding induces an algebraic 11-cycle on the Jacobian called the Ceresa cycle, which we denote by $\Cer(C,e)$. Although homologically trivial, Ceresa cycles of very general curves with genus ≥3\geq 3 are non-trivial under algebraic equivalence. It is generally unknown whether a very general curve admits a non-trivial Ceresa cycle modulo rational equivalence. In this paper, we explore the non-triviality under rational equivalence of the Ceresa cycle corresponding to the modular curve X0(N)X_0(N). By extending a method of Lupoian--Rawson involving computing the order of the shadow of the Atkin--Lehner involution wNw_N, we show $\Cer(X_0(kN),e)$ is non-trivial when N=57,58,77,82,85,88,91,93,111,112,115,123,133,135,141,143,145,147,155,161N = 57, 58, 77, 82, 85, 88, 91, 93, 111, 112, 115, 123, 133, 135, 141, 143, 145, 147, 155, 161 for k≥1k \geq 1. Moreover, we detail an algorithm for computing the shadow of the Hecke operator TℓT_\ell for odd primes ℓ\ell not dividing NN, which can be used to determine the non-triviality of the Ceresa cycle of X0(N)X_0(N).

Tools used

Magma
Magma

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Version history

  1. v1Submitted by Leonardo FinziInitial depositCurrentOct 20, 2026