Leonardo Finzi

math.AG — Algebraic Geometry

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

Contributed by Leonardo Finzi

For a curve $C$ and a fixed degree-one divisor $e$ of $C$, we can embed $C$ into its Jacobian. This embedding induces an algebraic $1$-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 $\geq 3$ 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 $X_0(N)$. By extending a method of Lupoian--Rawson involving computing the order of the shadow of the Atkin--Lehner involution $w_N$, we show $\Cer(X_0(kN),e)$ is non-trivial when $N = 57, 58, 77, 82, 85, 88, 91, 93, 111, 112, 115, 123, 133, 135, 141, 143, 145, 147, 155, 161$ for $k \geq 1$. Moreover, we detail an algorithm for computing the shadow of the Hecke operator $T_\ell$ for odd primes $\ell$ not dividing $N$, which can be used to determine the non-triviality of the Ceresa cycle of $X_0(N)$.