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

TL;DR

This paper proves that certain algebraic cycles associated with modular curves are non-torsion, by linking their properties to special cycles on orthogonal Shimura varieties and developing a new pullback formula.

Contribution

It establishes the non-torsion nature of Ceresa and Gross-Kudla-Schoen cycles for many modular curves, introducing a novel pullback formula for special divisors on modular curves.

Findings

Ceresa and Gross-Kudla-Schoen cycles are non-torsion for a large family of modular curves.

Develops a new pullback formula for special divisors on modular curves.

Relates cycle non-torsion to special cycles on orthogonal Shimura varieties.

Abstract

Associated to an algebraic curve $X$, there are two canonically constructed homologically trivial algebraic $1$-cycles, the Ceresa cycle in the Jacobian of $X$, and the Gross-Kudla-Schoen modified diagonal cycle in the triple product $X \times X \times X$. By a result of Shou-Wu Zhang, one is torsion if and only if the other is. In this paper, we prove that these two cycles associated to a large family of modular curves are non-torsion in the corresponding Chow groups. We obtain the result by relating this problem to the study of special cycles on orthogonal Shimura varieties. As the main ingredient and a result of independent interest, we develop a pullback formula for special divisors on modular curves embedded in their products via the diagonal map.