Non-torsion algebraic cycles on the Jacobians of Fermat quotients

TL;DR

This work studies the Abel-Jacobi images of Ceresa cycles $W_{k,e}-W_{k,e}^-$ on the Jacobians of Fermat quotient curves, proving non-torsion for all $k=1, obreak g-2$ under a number-theoretic condition: a prime divisor $p>7$ of $N$ with $p mid ab$ and $a^2+ab+b^2 ot ot o 0$ (mod $p$). The authors employ a mixed Hodge structure framework, via Harris-Pulte's formula and the extension class $E_e$, and relate the Abel-Jacobi image to iterated integrals, Künneth components, and explicit Jacobian points through Darmon-Rotger-Sols. A key step reduces to the $k=1$ case, then descends to a prime $p$ and to a quotient curve $C_p^{a,b}$ where a cyclic order-3 automorphism enables the construction of a non-torsion point $P_Z$ in the Jacobian; Gross-Rohrlich's non-torsion results then force the extension class to be nonzero, yielding non-torsion Ceresa cycles modulo rational equivalence. The paper therefore provides explicit non-torsion criteria for Ceresa cycles on a broad family of Fermat quotient curves and partially confirms related conjectures by Eskandari and Murty.

Abstract

We study the Abel-Jacobi image of the Ceresa cycle $W_{k, e}-W_{k, e}^-$, where $W_{k, e}$ is the image of the $k$th symmetric product of a curve $X$ with a base point $e$ on its Jacobian variety. For certain Fermat quotient curves of genus $g$, we prove that for any choice of the base point and $k \leq g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.

Theorems & Definitions (16)

• Theorem 1.1
• Lemma 2.1: cf. IS
• proof
• Lemma 2.3
• proof
• Theorem 2.4: GR
• Proposition 3.1
• proof
• Corollary 3.2
• proof